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BODY %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: genpin.dvi %%Pages: 34 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips genpin -o %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1996.03.26:1202 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N 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5385 3387 L 456 -103 V -456 571 R 228 -285 V 5385 3387 L 0 468 V 0 -468 R -229 286 V 229 182 V 0 -468 V -229 286 V 4928 3490 L 457 -103 V 4764 6356 M -122 152 V 122 0 R 0 -152 V -122 152 V -274 0 R -61 -49 V 457 -103 V -396 152 R -61 -49 V 0 49 V 0 -49 R -39 49 V 39 0 R 0 -49 V -39 49 V -179 0 R 218 -49 V 2187 3432 M 229 -285 V 228 182 V -457 103 V 457 -571 R -228 286 V 228 182 V 0 -468 V 0 468 R 228 -285 V 2644 2861 L 0 468 V 228 -285 V 229 183 V -457 102 V stroke grestore end showpage %%EndDocument @endspecial 1150 3354 a(Figure)g(1.)44 b(P)m(art)33 b(of)h(the)f(tiling)f Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(2\).)349 3564 y(These)40 b(tilings)e(all)g(arise)h(from)f(a)g (substitution)i(sc)m(heme)f(that)f(is)h(quite)f(similar)f(to)i(the)150 3683 y(pin)m(wheel)33 b(tiling)e(of)h(Con)m(w)m(a)m(y)g(and)g(Radin)g ([R1].)43 b(In)32 b(all)f(cases)i(the)f(tilings)f(ha)m(v)m(e)h(the)g (\\sibling)150 3803 y(edge-to-edge")41 b(prop)s(ert)m(y)-8 b(,)40 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b(of)h(or)-5 b(der)46 b Fi(n)f Fm(is)h(a)f(collection)g(of)h(tiles)f(that)g(is)g(similar)f (to)h Fi(T)3668 3641 y Fh(n)3722 3626 y Fm(.)150 3746 y(Equiv)-6 b(alen)m(tly)e(,)39 b(a)h(sup)s(ertile)f(of)h(order)g Fi(n)f Fm(is)h(a)f(region)h(of)f(the)h(form)f Fi(e)p Fm(\()p Fi(s)p Fm(\()p Fi(T)3013 3761 y Fh(n)3067 3746 y Fm(\)\),)h(where)h Fi(s)e Fm(is)g(a)150 3866 y(rescaling)33 b(and)h Fi(e)g Fm(is)f(a)g(Euclidean)h(motion.)150 4015 y Fn(Theorem)k(1:)47 b Fk(Given)35 b(any)i(right)f(triangle)g Fi(T)14 b Fk(,)36 b(ther)-5 b(e)36 b(exist)g(tilings)g(of)g(the)g (plane)g(with)h(right)150 4134 y(triangles)g(similar)h(to)g Fi(T)14 b Fk(,)38 b(such)f(that)h(any)g(\014nite)g(set)f(of)h(tiles)f (lies)g(in)h(a)g(sup)-5 b(ertile,)37 b(and)h(such)150 4254 y(that)e(the)f(ar)-5 b(e)g(as)35 b(of)g(tiles)g(ar)-5 b(e)36 b(b)-5 b(ounde)g(d)35 b(b)-5 b(oth)35 b(ab)-5 b(ove)35 b(and)h(b)-5 b(elow.)150 4403 y(Pr)g(o)g(of:)42 b Fm(First)30 b(pic)m(k)h(a)g(succession)i(of)e(in)m(tegers)h Fi(n)1993 4418 y Fl(1)2037 4403 y Fi(;)17 b(n)2142 4418 y Fl(2)2186 4403 y Fi(;)g(:)g(:)g(:)29 b Fm(and)i(tiles)g Fi(t)2814 4418 y Fh(i)2875 4403 y Fj(2)d Fi(T)3027 4418 y Fh(n)3076 4428 y Fg(i)3113 4403 y Fm(.)43 b(Let)31 b Fi(N)3440 4418 y Fl(1)3513 4403 y Fm(=)d Fi(n)3678 4418 y Fl(1)3722 4403 y Fm(.)150 4523 y(Pic)m(k)k(additional)f(in)m (tegers)i Fi(N)1289 4538 y Fh(i)1322 4523 y Fm(,)f Fi(i)c Fm(=)g(2)p Fi(;)17 b Fm(3)p Fi(;)g(:)g(:)g(:)m Fm(,)32 b(suc)m(h)h(that,)f(taking)f Fi(t)2722 4538 y Fh(i)2783 4523 y Fj(2)d Fi(T)2935 4538 y Fh(n)2984 4548 y Fg(i)3053 4523 y Fm(and)33 b(sub)s(dividing)150 4643 y Fi(T)208 4658 y Fh(n)257 4668 y Fg(i)331 4643 y Fm(an)38 b(additional)f Fi(N)1029 4658 y Fh(i)1087 4643 y Fj(\000)25 b Fi(n)1249 4658 y Fh(i)1320 4643 y Fm(times,)37 b Fi(t)1655 4658 y Fh(i)1723 4643 y Fj(\032)e Fi(T)1893 4658 y Fh(N)1956 4668 y Fg(i)2030 4643 y Fm(is)i(similar)f(to)h Fi(T)2650 4658 y Fh(N)2713 4668 y Fg(i)p Fd(\000)p Ff(1)2838 4643 y Fm(.)56 b(By)37 b(Lemma)f(3,)h(suc)m(h)150 4762 y(in)m(tegers)d(alw)m (a)m(ys)f(exist.)349 4912 y(No)m(w)g(pic)m(k)f(a)h(triangle)f(similar)f (to)h Fi(T)1739 4927 y Fl(0)1816 4912 y Fm(and)h(place)g(it)f(in)h(the) g(plane.)44 b(One)34 b(can)f(construct)150 5031 y(a)47 b(sup)s(ertile)g Fi(S)729 5046 y Fl(1)821 5031 y Fm(of)g(order)g Fi(N)1302 5046 y Fl(1)1394 5031 y Fm(that)g(con)m(tains)g(this)g (triangle)f(in)h(p)s(osition)g Fi(t)3175 5046 y Fl(1)3219 5031 y Fm(.)85 b(One)48 b(then)150 5151 y(constructs)42 b(a)e(sup)s(ertile)h Fi(S)1201 5166 y Fl(2)1286 5151 y Fm(of)g(order)g Fi(N)1755 5166 y Fl(2)1840 5151 y Fm(suc)m(h)h(that)f Fi(S)2357 5166 y Fl(1)2442 5151 y Fm(sits)g(inside)g Fi(S)2979 5166 y Fl(2)3064 5151 y Fm(as)g Fi(t)3230 5166 y Fl(2)3315 5151 y Fm(sits)f(inside)150 5270 y Fi(T)208 5285 y Fh(n)257 5295 y Ff(2)301 5270 y Fm(.)72 b(One)43 b(con)m(tin)m(ues)h(the)f(pro)s(cess,)j(building)c(sup)s(ertile)h Fi(S)2509 5285 y Fh(k)r Fl(+1)2701 5270 y Fm(suc)m(h)h(that)e Fi(S)3221 5285 y Fh(k)3313 5270 y Fm(sits)g(inside)1925 5519 y(6)p eop %%Page: 7 7 7 6 bop 150 100 a Fi(S)211 115 y Fh(k)r Fl(+1)394 100 y Fm(as)34 b Fi(t)553 115 y Fh(k)r Fl(+1)736 100 y Fm(sits)f(inside)h Fi(T)1255 115 y Fh(n)1304 126 y Fg(k)q Ff(+1)1438 100 y Fm(.)349 249 y(The)k(union)g(of)g(all)f(the)h(sup)s(ertiles)g(is)f(a) h(tiling)e(of)i(an)f(in\014nite)h(region.)57 b(F)-8 b(or)37 b(almost)f(all)150 369 y(c)m(hoices)43 b(of)f(the)g Fi(t)832 384 y Fh(i)865 369 y Fm('s)g(\(e.g.,)h(ha)m(ving)e(the)h(edges)h(of)f Fi(t)2176 384 y Fh(i)2251 369 y Fm(lie)f(in)h(the)g(in)m(terior)g(of)g Fi(T)3238 384 y Fh(n)3287 394 y Fg(i)3365 369 y Fm(in\014nitely)150 488 y(often\),)33 b(this)h(region)f(will)f(b)s(e)h(the)h(en)m(tire)g (plane.)349 638 y(Since)51 b(the)f(ratio)e(of)i(largest)f(to)g (smallest)g(triangle)g(is)g(uniformly)g(b)s(ounded)i(for)f Fi(S)3696 653 y Fh(n)150 757 y Fm(b)m(y)42 b(Lemma)f(1,)j(no)f(tile)e (ma)m(y)g(ha)m(v)m(e)i(h)m(yp)s(oten)m(use)h(longer)e(than)h(max)n(\()p Fi(c=a;)17 b Fm(2)p Fi(c=b)p Fm(\))40 b(times)i(the)150 877 y(h)m(yp)s(oten)m(use)30 b(of)f(a)f(\014xed)h(tile,)f(and)h(no)f (tile)g(ma)m(y)f(ha)m(v)m(e)i(h)m(yp)s(oten)m(use)h(less)f(than)f(min)o (\()p Fi(a=c;)17 b(b=)p Fm(2)p Fi(c)p Fm(\))150 996 y(times)29 b(the)h(h)m(yp)s(oten)m(use)h(of)f(the)h(same)e(\014xed)h(tile.)42 b(This)30 b(pro)m(vides)h(b)s(oth)f(an)g(upp)s(er)g(and)h(lo)m(w)m(er) 150 1116 y(b)s(ound)j(to)f(the)h(size)f(of)h(the)f(tiles.)p 1534 1126 60 80 v 349 1265 a(In)g(this)f(construction,)h(man)m(y)f(c)m (hoices)h(w)m(ere)g(made.)43 b(Di\013eren)m(t)34 b(c)m(hoices)f (generally)f(lead)150 1385 y(to)46 b(di\013eren)m(t)h(tilings,)i(but)e (these)g(di\013eren)m(t)h(tilings)d(ha)m(v)m(e)i(man)m(y)f(prop)s (erties)h(in)f(common.)150 1504 y(The)38 b(follo)m(wing)e(Theorems)h(2) g(and)h(3)f(apply)g(to)g(all)f(tilings)g(constructed)j(in)e(the)h (manner)f(of)150 1624 y(Theorem)c(1.)150 1773 y Fn(Theorem)49 b(2:)66 b Fk(In)45 b(a)g(tiling,)i(the)e(numb)-5 b(er)46 b(of)e(orientations)g(in)h(which)f(the)h(b)-5 b(asic)44 b(triangle)150 1893 y(app)-5 b(e)g(ars)35 b(is)g(\014nite)g(if)g Fi(\022)s(=\031)j Fk(is)d(r)-5 b(ational)35 b(and)h(in\014nite)e(if)h Fi(\022)s(=\031)k Fk(is)c(irr)-5 b(ational.)150 2042 y(Pr)g(o)g(of:)50 b Fm(First)37 b(supp)s(ose)h(that)f Fi(\022)s(=\031)i Fm(is)e(rational.)54 b(Consider)37 b(a)g(tile,)g(p)s(ositioned)g(as)g(in)g(Figure)150 2162 y(3,)i(with)e(side)i Fi(b)f Fm(along)f(the)i Fi(x)f Fm(axis.)57 b(Let)39 b Fi(P)52 b Fm(denote)38 b(re\015ection)h(ab)s(out)f(the)h Fi(x)e Fm(axis,)i(and)f(let)150 2281 y Fi(R)226 2296 y Fh(\013)328 2281 y Fm(denote)46 b(a)g(coun)m(terclo)s(c)m(kwise)g (rotation)f(b)m(y)g(angle)h Fi(\013)p Fm(.)81 b(After)46 b(sub)s(dividing)g(once,)j(the)150 2401 y(orien)m(tations)38 b(of)g(the)g(\014v)m(e)g(daugh)m(ter)h(tiles,)f(relativ)m(e)f(to)g(the) i(paren)m(t)f(tile,)g(are)g(giv)m(en)g(b)m(y)g(the)150 2521 y(follo)m(wing)26 b(elemen)m(ts)h(of)g Fi(O)s Fm(\(2\):)40 b Fi(R)1423 2536 y Fh(\022)1468 2521 y Fi(P)g Fm(\(t)m(wice\),)27 b Fi(R)2005 2536 y Fh(\022)2050 2521 y Fm(,)h Fi(R)2182 2536 y Fh(\031)s Fl(+)p Fh(\022)2337 2521 y Fm(,)g(and)f Fi(R)2656 2539 y Fh(\031)s(=)p Fl(2+)p Fh(\022)2891 2521 y Fi(P)14 b Fm(.)42 b(The)27 b(orien)m(tations)150 2640 y(of)35 b(tiles)f(in)g(a)g(further)i(sub)s(division)e(are)h(w)m(ords)g (in)f(these)i(\014v)m(e)e(elemen)m(ts)h(of)f Fi(O)s Fm(\(2\).)47 b(Ho)m(w)m(ev)m(er,)150 2760 y(with)33 b Fi(\022)s(=\031)i Fm(rational,)d(these)i(\014v)m(e)f(elemen)m(ts)g(generate)h(a)e (\014nite)i(subgroup)g(of)f Fi(O)s Fm(\(2\),)f(so)h(only)f(a)150 2879 y(\014nite)i(n)m(um)m(b)s(er)f(of)h(orien)m(tations)f(can)h(ev)m (er)f(app)s(ear)h(in)f(a)g(future)h(sub)s(division.)349 3029 y(Since)i(an)m(y)e(region)g(of)h(our)g(tiling)e(of)i(the)f(plane)h (sits)g(inside)g(a)f(sup)s(ertile,)h(an)m(y)f(t)m(w)m(o)g(tiles)150 3148 y(m)m(ust)42 b(ha)m(v)m(e)h(their)g(orien)m(tations,)h(relativ)m (e)e(to)g(the)h(sup)s(ertile)g(itself,)i(in)e(this)f(group.)73 b(Th)m(us)150 3268 y(their)37 b(orien)m(tations,)g(relativ)m(e)g(to)f (eac)m(h)i(other,)g(and)g(hence)g(to)f(a)g(\014xed)g(reference)i(tile,) e(m)m(ust)150 3387 y(lie)31 b(in)g(the)h(group.)44 b(Th)m(us)32 b(only)f(a)g(\014nite)g(n)m(um)m(b)s(er)h(of)g(orien)m(tations)f(can)h (app)s(ear)f(in)h(the)f(tiling.)349 3537 y(No)m(w)k(supp)s(ose)h(that)e Fi(\022)s(=\031)k Fm(is)d(irrational.)47 b(In)35 b(a)g(basic)g(sub)s (division,)g(w)m(e)g(will)f(k)m(eep)h(trac)m(k)150 3656 y(only)43 b(of)g(the)g(4)g(triangles)g(of)h(h)m(yp)s(oten)m(use)g Fi(b=)p Fm(2,)h(ignoring)e(the)g(triangle)g(of)g(h)m(yp)s(oten)m(use)i Fi(a)p Fm(.)150 3776 y(When)35 b(these)f(4)f(triangles)f(divide,)h(w)m (e)h(will)e(only)g(k)m(eep)i(trac)m(k)f(of)g(the)h(16)f(resultan)m(t)g (triangles)150 3895 y(of)h(side)g Fi(b)507 3859 y Fl(2)551 3895 y Fi(=)p Fm(4)p Fi(c)p Fm(,)f(and)h(so)f(on.)44 b(In)34 b(the)g(second)h(generation)e(w)m(e)h(\014nd)g(orien)m(tations) f(1)28 b(=)g(\()p Fi(R)3544 3910 y Fh(\022)3589 3895 y Fi(P)14 b Fm(\))3706 3859 y Fl(2)150 4015 y Fm(and)29 b Fi(R)415 4030 y Fl(2)p Fh(\022)527 4015 y Fm(=)f Fi(R)709 3979 y Fl(2)708 4043 y Fh(\022)753 4015 y Fm(,)h(among)e(others.)43 b(In)28 b(the)h(2)p Fi(n)p Fm(-th)f(generation)g(w)m(e)h(\014nd)g(1)p Fi(;)17 b(R)3003 4030 y Fl(2)p Fh(\022)3087 4015 y Fi(;)g(R)3208 4030 y Fl(4)p Fh(\022)3291 4015 y Fi(;)g(:)g(:)g(:)d(;)j(R)3589 4030 y Fl(2)p Fh(n\022)3722 4015 y Fm(.)150 4134 y(Since)31 b Fi(\022)s(=\031)i Fm(is)d(irrational,)f(these)i(2)p Fi(n)16 b Fm(+)g(1)30 b(orien)m(tations)g(are)h(distinct.)43 b(Since)31 b(our)f(tiling)f(of)i(the)150 4254 y(plane)k(con)m(tains)g (sup)s(ertiles)h(of)f(arbitrarily)e(large)h(size,)h(there)h(is)f(no)f (b)s(ound)i(to)e(the)h(n)m(um)m(b)s(er)150 4374 y(of)e(di\013eren)m(t)i (orien)m(tations)e(that)g(app)s(ear.)p 1847 4384 V 150 4523 a Fn(Theorem)46 b(3:)59 b Fk(In)42 b(a)h(tiling,)g(the)f(numb)-5 b(er)43 b(of)e(sizes)g(in)h(which)g(the)g(b)-5 b(asic)41 b(triangle)h(app)-5 b(e)g(ars)150 4643 y(is)32 b(in\014nite)f(if)g Fm(ln)q(\(sin\()p Fi(\022)s Fm(\)\))p Fi(=)17 b Fm(ln)n([cos\()p Fi(\022)s Fm(\))p Fi(=)p Fm(2])31 b Fk(is)g(irr)-5 b(ational)32 b(and)g(\014nite)g(if)f Fm(ln\(sin\()p Fi(\022)s Fm(\)\))p Fi(=)17 b Fm(ln)o([cos\()p Fi(\022)s Fm(\))p Fi(=)p Fm(2])150 4762 y Fk(is)40 b(r)-5 b(ational.)60 b(In)41 b(p)-5 b(articular,)41 b(if)f Fm(ln\(sin\()p Fi(\022)s Fm(\)\))p Fi(=)17 b Fm(ln)n([cos\()p Fi(\022)s Fm(\))p Fi(=)p Fm(2])35 b(=)i Fi(p=q)t Fk(,)42 b(with)e Fi(p)h Fk(and)f Fi(q)45 b Fk(r)-5 b(elatively)150 4882 y(prime)35 b(inte)-5 b(gers,)34 b(then)i(the)f(numb)-5 b(er)37 b(of)e(sizes)f(in)h(the)h(tiling)f(is)g Fm(max)n(\()p Fi(p;)17 b(q)t Fm(\))p Fk(.)150 5031 y(Pr)-5 b(o)g(of:)54 b Fm(Let)39 b Fi(A)d Fm(=)h Fi(a=c)i Fm(and)g(let)f Fi(B)j Fm(=)d Fi(b=)p Fm(2)p Fi(c)p Fm(.)60 b(If)39 b(ln\()p Fi(A)p Fm(\))p Fi(=)17 b Fm(ln)n(\()p Fi(B)5 b Fm(\))38 b(is)h(irrational,)f(the)h(only)f(w)m(a)m(y)150 5151 y(t)m(w)m(o)k(monomials)e Fi(A)930 5115 y Fh(a)978 5151 y Fi(B)1059 5115 y Fh(b)1140 5151 y Fm(and)j Fi(A)1418 5115 y Fh(c)1458 5151 y Fi(B)1539 5115 y Fh(d)1627 5151 y Fm(can)g(equal)f(is)g(if)g Fi(a)g Fm(=)h Fi(c)f Fm(and)h Fi(b)f Fm(=)h Fi(d)p Fm(.)71 b(W)-8 b(e)42 b(will)f(sho)m(w)150 5270 y(that)c(the)h(sizes)g(of)g(triangles)f(in)g Fi(T)1483 5285 y Fh(n)1575 5270 y Fm(\(relativ)m(e)g(to)g(the)g(original)g (triangle\))f(is)i(giv)m(en)f(b)m(y)g(suc)m(h)1925 5519 y(7)p eop %%Page: 8 8 8 7 bop 150 100 a Fm(monomials,)32 b(and)j(that)f(the)h(n)m(um)m(b)s (er)g(of)g(distinct)g(p)s(o)m(w)m(ers)g(of)g Fi(A)f Fm(gro)m(ws)h (without)f(b)s(ounds)i(as)150 219 y Fi(n)g Fj(!)f(1)p Fm(.)59 b(This)38 b(will)f(sho)m(w)i(that)f(the)g(n)m(um)m(b)s(er)h(of) g(distinct)f(sizes)h(gro)m(ws)f(without)g(b)s(ound)h(a)150 339 y Fi(n)27 b Fj(!)h(1)p Fm(.)349 488 y(In)40 b(eac)m(h)h(sub)s (division)e(there)i(are)e(4)h(tiles)f(of)h(size)g Fi(B)k Fm(relativ)m(e)39 b(to)g(the)h(paren)m(t)g(and)h(one)150 608 y(tile)28 b(of)h(size)g Fi(A)p Fm(.)43 b(Th)m(us)30 b(the)f(descendan)m(ts)i(of)e(a)g(giv)m(en)f(tile)g(all)g(ha)m(v)m(e)i (sizes)f(that)g(are)g(monomials)150 727 y Fi(A)225 691 y Fh(a)273 727 y Fi(B)354 691 y Fh(b)430 727 y Fm(relativ)m(e)35 b(to)h(the)g(ancestor.)54 b(F)-8 b(or)36 b(ev)m(ery)g Fi(n)c(>)i Fm(0,)i Fi(T)2351 742 y Fh(n)2442 727 y Fm(con)m(tains)g(at) g(least)g(one)h(tile)e(with)150 847 y(size)h Fi(A)416 811 y Fl(0)460 847 y Fi(B)541 811 y Fh(b)581 847 y Fm(;)g(just)h(tak)m (e)e(a)g Fi(B)40 b Fm(c)m(hild)c(of)g(a)f Fi(B)40 b Fm(c)m(hild)d(of)e Fi(:)17 b(:)g(:)34 b Fm(of)i(one)g(of)g(the)f(original)g Fi(B)40 b Fm(c)m(hildren)150 966 y(\(or)g(the)g Fi(B)45 b Fm(c)m(hild)c(itself,)h(if)e(it)g(has)g(not)h(sub)s(divided\).)65 b(F)-8 b(or)40 b(ev)m(ery)g Fi(n)g(>)f Fm(0,)j Fi(T)3169 981 y Fh(n)3264 966 y Fm(con)m(tains)e(at)150 1086 y(least)f(one)h (tile)f(with)g(size)h Fi(A)1259 1050 y Fl(1)1303 1086 y Fi(B)1384 1050 y Fh(b)1424 1086 y Fm(;)i(tak)m(e)d(a)g Fi(B)5 b(B)g(B)20 b(:)d(:)g(:)38 b Fm(descendan)m(t)j(of)f(the)g (original)e Fi(A)h Fm(c)m(hild.)150 1205 y(Once)44 b Fi(n)e Fm(is)h(large)f(enough)h(to)f(ha)m(v)m(e)h(the)g(original)e Fi(A)h Fm(c)m(hild)h(divide,)i(there)e(is)f(at)g(least)h(one)150 1325 y(tile)37 b(with)f(size)i Fi(A)824 1289 y Fl(2)869 1325 y Fi(B)950 1289 y Fh(b)989 1325 y Fm(.)56 b(In)38 b(general,)g(once)g Fi(n)f Fm(is)g(large)f(enough)j(to)d(allo)m(w)h(a)g Fi(a)p Fm(-th)g(generation)150 1445 y Fi(AA)17 b(:)g(:)g(:)29 b Fm(c)m(hild,)j(it)e(will)h(alw)m(a)m(ys)g(ha)m(v)m(e)h(at)f(least)g (one)h(tile)e(whose)i(size)g(has)g(exactly)e Fi(a)h Fm(p)s(o)m(w)m(ers) i(of)150 1564 y Fi(A)p Fm(.)44 b(This)33 b(completes)g(the)h (irrational)d(case.)349 1714 y(No)m(w)42 b(supp)s(ose)i(that)d(ln\()p Fi(A)p Fm(\))p Fi(=)17 b Fm(ln)o(\()p Fi(B)5 b Fm(\))42 b(=)h Fi(p=q)t Fm(,)g(with)f Fi(p)h Fm(and)f Fi(q)k Fm(relativ)m(ely)41 b(prime.)70 b(Th)m(us)150 1833 y Fi(A)225 1797 y Fh(q)296 1833 y Fm(=)29 b Fi(B)483 1797 y Fh(p)528 1833 y Fm(.)43 b(Let)29 b Fi(r)i Fm(=)d Fi(A)1029 1797 y Fl(1)p Fh(=p)1183 1833 y Fm(=)g Fi(B)1369 1797 y Fl(1)p Fh(=q)1493 1833 y Fm(.)43 b(Ev)m(ery)29 b(monomial)d Fi(A)2377 1797 y Fh(a)2425 1833 y Fi(B)2506 1797 y Fh(b)2575 1833 y Fm(is)j(a)g(p)s(o)m (w)m(er)h(of)g Fi(r)s Fm(.)42 b(Assume)30 b(for)150 1953 y(the)36 b(momen)m(t)d(that)i Fi(p)d Fj(\024)f Fi(q)t Fm(.)51 b(By)34 b(Lemma)g(1,)h(the)h(ratio)e(of)i(sizes)g(of)f(an)m(y)h (t)m(w)m(o)f(tiles)g(is)g(greater)150 2072 y(than)e Fi(B)g Fm(=)28 b Fi(r)644 2036 y Fh(q)687 2072 y Fm(.)44 b(Th)m(us)35 b(only)d(at)h(most)f Fi(q)37 b Fm(distinct)c(sizes)h(can)g(app)s(ear.) 44 b(T)-8 b(o)32 b(see)i(that)f Fi(q)k Fm(sizes)d Fk(do)150 2192 y Fm(app)s(ear,)f(w)m(e)h(note)f(that)g Fi(A)1166 2156 y Fl(0)1211 2192 y Fi(B)1292 2156 y Fh(b)1327 2166 y Ff(0)1370 2192 y Fi(;)17 b(A)1490 2156 y Fl(1)1534 2192 y Fi(B)1615 2156 y Fh(b)1650 2166 y Ff(1)1693 2192 y Fi(;)g(:)g(:)g(:)d(;)j(A)1990 2156 y Fh(q)s Fe(\000)p Fl(1)2135 2192 y Fi(B)2216 2156 y Fh(b)2251 2166 y Fg(q)r Fd(\000)p Ff(1)2417 2192 y Fm(are)33 b(all)f(distinct)i(p)s(o)m(w)m (ers)g(of)f Fi(r)s Fm(.)349 2341 y(If)41 b Fi(p)f(>)g(q)t Fm(,)i(Lemma)d(1)i(states)f(that)g(the)h(ratio)f(of)h(an)m(y)f(t)m(w)m (o)h(sizes)g(is)f(at)g(least)h Fi(A)e Fm(=)i Fi(r)3677 2305 y Fh(p)3722 2341 y Fm(,)150 2461 y(so)d(at)g(most)f Fi(p)h Fm(di\013eren)m(t)h(sizes)g(can)f(o)s(ccur.)59 b(W)-8 b(e)39 b(pro)s(duce)g Fi(p)f Fm(di\013eren)m(t)h(sizes)g(b)m(y)f (examining)150 2580 y(di\013eren)m(t)c(p)s(o)m(w)m(ers)g(of)g Fi(B)5 b Fm(.)43 b(In)34 b(either)f(case,)h(the)g(n)m(um)m(b)s(er)f(of) h(distinct)f(sizes)h(is)f(max)o(\()p Fi(p;)17 b(q)t Fm(\).)p 3658 2590 60 80 v 349 2730 a(W)-8 b(e)28 b(refer)g(to)f(tilings)f(with) h(ln\()p Fi(A)p Fm(\))p Fi(=)17 b Fm(ln)o(\()p Fi(B)5 b Fm(\))26 b(=)i Fi(p=q)k Fm(as)27 b(\()p Fi(p=q)t Fm(\))i Fk(r)-5 b(ational)30 b(tilings)p Fm(,)e(and)f(denote)150 2849 y(the)38 b(class)g(of)f(suc)m(h)i(tilings)e(as)h Fi(T)14 b(il)r Fm(\()p Fi(p=q)t Fm(\).)56 b(F)-8 b(or)37 b Fi(z)42 b Fm(irrational,)37 b(w)m(e)h(will)e(similarly)f(denote)k (the)150 2969 y(class)26 b(of)h(tilings)e(with)h(ln\()p Fi(A)p Fm(\))p Fi(=)17 b Fm(ln)o(\()p Fi(B)5 b Fm(\))26 b(=)i Fi(z)j Fm(as)26 b Fi(T)14 b(il)r Fm(\()p Fi(z)t Fm(\).)42 b(The)27 b(di\013eren)m(t)g(tilings)e(in)h(a)g(class)h(are)f (all)150 3088 y(deriv)m(ed)36 b(from)f(the)g(same)g(substitution)h (rule,)f(and)h(ha)m(v)m(e)g(man)m(y)e(prop)s(erties)i(deriv)-6 b(able)36 b(from)150 3208 y(this)43 b(rule.)75 b(When)44 b(discussing)h(suc)m(h)g(prop)s(erties,)h(w)m(e)e(will)e(sometimes)f (refer)k(to)e(a)g(t)m(ypical)150 3328 y(elemen)m(t)33 b(of)h(the)f(class)h(as)f(\\the)g(tiling)g Fi(T)14 b(il)r Fm(\()p Fi(z)t Fm(\)".)349 3477 y(Note)39 b(that)g(ln\(sin\()p Fi(\022)s Fm(\)\))p Fi(=)17 b Fm(ln)n([cos\()p Fi(\022)s Fm(\))p Fi(=)p Fm(2])38 b(is)h(a)g(strictly)f(decreasing)j(function)f (of)f Fi(\022)j Fm(on)e(the)150 3597 y(in)m(terv)-6 b(al)25 b(\(0)p Fi(;)17 b(\031)t(=)p Fm(2\).)39 b(F)-8 b(rom)23 b(this)i(monotonicit)m(y)-8 b(,)24 b(and)i(from)e(the)h(coun)m(tabilit) m(y)g(of)g(the)g(rationals,)150 3716 y(it)47 b(is)h(clear)g(that)g (only)g(a)f(coun)m(table)i(set)g(of)f(angles)g Fi(\022)j Fm(giv)m(e)c(rise)h(to)g(a)g(\014nite)g(n)m(um)m(b)s(er)h(of)150 3836 y(rotations,)33 b(and)h(only)f(a)g(coun)m(table)i(set)f(of)g (angles)f Fi(\022)k Fm(giv)m(e)c(rise)h(to)f(a)g(\014nite)h(n)m(um)m(b) s(er)h(of)e(sizes.)150 3955 y(The)h(in)m(tersection)g(of)f(these)h(t)m (w)m(o)g(coun)m(table)g(sets)f(turns)h(out)g(to)e(b)s(e)i(a)f(single)g (p)s(oin)m(t.)150 4105 y Fn(Theorem)k(4:)44 b Fk(The)35 b(only)g(angle)g(that)g(gives)f(rise)g(to)h(b)-5 b(oth)34 b(a)h(\014nite)g(numb)-5 b(er)35 b(of)f(orientations)150 4224 y(and)h(a)h(\014nite)f(numb)-5 b(er)36 b(of)f(sizes)g(is)g Fi(\022)30 b Fm(=)e Fi(\031)t(=)p Fm(4)p Fk(.)44 b(That)36 b(is,)f(the)g(tiling)g Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(3\))p Fk(.)150 4374 y(Pr)-5 b(o)g(of:)60 b Fm(Let)42 b Fi(x)g Fm(=)h(exp\()p Fi(i\022)s Fm(\),)g(with)f(0)g Fi(<)h(\022)h(<)f(\031)t(=)p Fm(2.)69 b(W)-8 b(e)43 b(are)f(lo)s(oking) e(for)i(solutions)g(to)g(the)150 4493 y(equation)33 b(sin\()p Fi(\022)s Fm(\))807 4457 y Fh(q)878 4493 y Fm(=)28 b([cos\()p Fi(\022)s Fm(\))p Fi(=)p Fm(2])1400 4457 y Fh(p)1444 4493 y Fm(,)33 b(whic)m(h)h(w)m(e)g(rewrite)f(as)1280 4692 y(2)1330 4651 y Fh(q)1374 4692 y Fm(\()p Fi(x)22 b Fm(+)29 b(\026)-56 b Fi(x)o Fm(\))1687 4651 y Fh(p)1761 4692 y Fm(=)28 b(2)1916 4651 y Fl(2)p Fh(p)2002 4692 y Fm(\()p Fj(\000)p Fi(i)p Fm(\))2191 4651 y Fh(q)2235 4692 y Fm(\()p Fi(x)22 b Fj(\000)29 b Fm(\026)-57 b Fi(x)p Fm(\))2548 4651 y Fh(q)2592 4692 y Fi(;)925 b Fm(\(2)p Fi(:)p Fm(1\))150 4891 y(where)41 b(\026)-56 b Fi(x)29 b Fm(=)g(exp\()p Fj(\000)p Fi(i\022)s Fm(\))g(=)h Fi(x)1215 4855 y Fe(\000)p Fl(1)1322 4891 y Fm(.)46 b(Note)34 b(that,)f(for)h (\014xed)h Fi(p)p Fm(,)f Fi(q)t Fm(,)g(there)g(is)g(at)g(most)f(one)h (solution)150 5011 y(to)f(equation)g(\(1\))f(in)h(the)h(\014rst)g (quadran)m(t,)f(since)h(ln[sin\()p Fi(\022)s Fm(\)])p Fi(=)17 b Fm(ln)n([cos\()p Fi(\022)s Fm(\))p Fi(=)p Fm(2])31 b(is)j(monotonic.)349 5160 y(Since)g(b)m(y)g(assumption)f Fi(\022)i Fm(is)e(a)g(rational)g(m)m(ultiple)f(of)h Fi(\031)t Fm(,)f Fi(x)i Fm(is)f(a)g(primitiv)m(e)e Fi(n)p Fm(-th)i(ro)s(ot)g(of) 150 5280 y(unit)m(y)27 b(for)f(some)g(in)m(teger)h Fi(n)p Fm(.)42 b(If)27 b Fi(x)f Fm(is)h(a)f(solution)h(and)g Fi(q)j Fm(is)d(ev)m(en,)h(then)g(equation)e(\(1\))g(has)h(\(real\))1925 5519 y(8)p eop %%Page: 9 9 9 8 bop 150 100 a Fm(in)m(teger)33 b(co)s(e\016cien)m(ts,)g(and)g Fk(al)5 b(l)34 b Fm(the)f(primitiv)m(e)e Fi(n)p Fm(-th)h(ro)s(ots)g(of) h(unit)m(y)f(are)h(also)f(solutions.)44 b(If)33 b Fi(q)150 219 y Fm(is)j(o)s(dd,)i(all)d(the)i(primitiv)m(e)e Fi(n)p Fm(-th)h(ro)s(ots)g(of)h(unit)m(y)f(are)h(solutions)f(either)h(to)f (equation)g(\(1\))g(or)150 339 y(to)d(the)g(conjugate)h(equation)1358 583 y(2)1408 542 y Fh(q)1452 583 y Fm(\()p Fi(x)21 b Fm(+)29 b(\026)-56 b Fi(x)p Fm(\))1765 542 y Fh(p)1838 583 y Fm(=)29 b(2)1994 542 y Fl(2)p Fh(p)2079 583 y Fi(i)2113 542 y Fh(q)2157 583 y Fm(\()p Fi(x)22 b Fj(\000)29 b Fm(\026)-56 b Fi(x)p Fm(\))2471 542 y Fh(q)2515 583 y Fi(:)1002 b Fm(\(2)p Fi(:)p Fm(2\))150 827 y(Equation)34 b(\(2\),)f(with)h Fi(q)39 b Fm(o)s(dd,)34 b(has)h(no)g(solutions)f(in)h (the)f(\014rst)h(quadran)m(t,)g(as)f(the)h(righ)m(t)f(hand)150 946 y(side)39 b(is)f(p)s(ositiv)m(e)g(but)h(the)f(left)h(hand)g(side)g (is)f(negativ)m(e.)59 b(Since)40 b(equation)e(\(1\))f(admits)h(only)150 1066 y(one)33 b(solution)f(in)g(the)h(\014rst)g(quadran)m(t,)g(there)g (m)m(ust)f(b)s(e)h(exactly)e(one)i(primitiv)m(e)e Fi(n)p Fm(-th)h(ro)s(ot)g(of)150 1185 y(unit)m(y)h(in)g(the)h(\014rst)g (quadran)m(t.)349 1335 y(This)54 b(means)f(that)g Fi(n)g Fm(m)m(ust)g(equal)g(5,)58 b(6,)f(7,)h(8,)g(10,)f(12)c(or)g(18.)104 b(Chec)m(king)54 b(these)150 1454 y(individually)-8 b(,)56 b(w)m(e)e(see)g(that)f(only)g Fi(n)61 b Fm(=)h(8,)57 b(or)d Fi(\022)63 b Fm(=)f Fi(\031)t(=)p Fm(4,)57 b(yields)c(a)g (rational)f(v)-6 b(alue)54 b(of)150 1574 y(ln[sin\()p Fi(\022)s Fm(\)])p Fi(=)17 b Fm(ln)n([cos\()p Fi(\022)s Fm(\))p Fi(=)p Fm(2].)73 b(In)44 b(that)f(one)h(case)g(sin\()p Fi(\022)s Fm(\))g(=)h(cos)q(\()p Fi(\022)s Fm(\))e(=)2824 1491 y Fj(p)p 2907 1491 50 4 v 83 x Fm(2)o Fi(=)p Fm(2,)j(and)e(sin) 3456 1532 y Fl(3)3501 1574 y Fm(\()p Fi(\022)s Fm(\))g(=)150 1693 y(cos\()p Fi(\022)s Fm(\))p Fi(=)p Fm(2.)p 642 1703 60 80 v 349 1843 a(In)31 b(a)g(p)s(erio)s(dic)g(tiling,)f(all)g(of)h (the)g(sizes)g(and)h(orien)m(tations)e(are)h(exhibited)g(in)g(a)g (compact)150 1962 y(region,)d(so)f(neither)h(the)f(n)m(um)m(b)s(er)h (of)f(sizes)h(nor)g(the)f(n)m(um)m(b)s(er)h(of)f(orien)m(tations)g(can) g(b)s(e)h(in\014nite.)150 2082 y(Th)m(us)34 b(w)m(e)g(ha)m(v)m(e)150 2231 y Fn(Corollary)g Fk(If)h Fi(z)e Fj(6)p Fm(=)28 b(1)p Fi(=)p Fm(3)p Fk(,)34 b(then)i(the)f(tiling)h Fi(T)14 b(il)r Fm(\()p Fi(z)t Fm(\))36 b Fk(is)e(not)i(p)-5 b(erio)g(dic)p Fm(.)349 2381 y(In)28 b(fact,)h(it)e(will)g(turn)h(out)f(that)h Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(3\))26 b(is)i(not)g(p)s(erio)s(dic,)g (either.)42 b(This)28 b(will)f(b)s(e)h(sho)m(wn)150 2500 y(in)33 b Fj(x)q Fm(6.)150 2650 y Fj(x)p Fn(3.)51 b(An)38 b(example)f(with)g(t)m(w)m(o)g(sizes.)349 2799 y Fm(In)45 b(this)g(section)g(w)m(e)h(consider)g(in)e(detail)h(the)g(tiling)f Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(2\),)46 b(sho)m(wn)g(in)f(Figure)g (1.)150 2919 y(This)e(example)f(is)h(c)m(hosen)i(not)e(as)g(a)g(sp)s (ecial)g(case,)j(but)d(rather)g(as)h(a)e(simple)h(example)f(of)150 3038 y(some)j(general)h(phenomena.)81 b(Based)46 b(on)f(the)h (statistical)e(analysis)h(of)h Fj(x)p Fm(4,)i(w)m(e)e(exp)s(ect)g(all) 150 3158 y(tilings)34 b Fi(T)14 b(il)r Fm(\()p Fi(p=q)t Fm(\),)34 b(with)g Fi(q)g(>)d Fm(1)j(and)i Fi(p=q)e Fj(6)p Fm(=)c(1)p Fi(=)p Fm(3,)k(to)g(b)s(e)h(qualitativ)m(ely)d(similar)h(to) h Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(2\).)150 3278 y(Sp)s(eci\014cally) -8 b(,)41 b(in)f(all)g(these)h(cases)g(the)f(p)s(opulation)g(matrix)e (has)j(t)m(w)m(o)f(or)g(more)f(eigen)m(v)-6 b(alues)150 3397 y(with)34 b(mo)s(dulus)g(bigger)g(than)h(one.)47 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(triangles.)43 b(Similarly)31 b(for)j(all)e Fi(T)1840 3594 y Fl(2)p Fh(n)1934 3579 y Fm(.)349 3728 y(The)47 b(ev)m(olution)f(of)h(these)g(legs)f(and)h(h)m(yp)s(oten)m(uses)h(is)f (a)f(substitution)g(system)g(in)g(its)150 3848 y(o)m(wn)37 b(righ)m(t,)h(only)e(in)i(one)f(dimension.)55 b(There)38 b(are)g(four)f(sym)m(b)s(ols,)g Fi(H)2863 3811 y Fl(+)2929 3848 y Fm(,)h Fi(H)3086 3811 y Fe(\000)3153 3848 y Fm(,)g Fi(L)3287 3811 y Fl(+)3390 3848 y Fm(and)f Fi(L)3655 3811 y Fe(\000)3722 3848 y Fm(,)150 3967 y(represen)m(ting)28 b(the)f(t)m(w)m(o)f(orien)m(tations)g(of)h(the)g(h)m(yp)s(oten)m(use)h (and)f(long)f(leg,)h(resp)s(ectiv)m(ely)-8 b(.)41 b(F)-8 b(rom)150 4087 y(Figure)33 b(6b,)g(w)m(e)h(see)g(that)f(the)h (substitution)f(rule,)g(whic)m(h)h(w)m(e)g(denote)g Fi(\033)2917 4102 y Fl(0)2962 4087 y Fm(,)f(is)312 4330 y Fi(\033)369 4345 y Fl(0)414 4330 y Fm(\()p Fi(H)544 4289 y Fl(+)610 4330 y Fm(\))27 b(=)h Fi(L)849 4289 y Fl(+)915 4330 y Fi(L)983 4289 y Fe(\000)1050 4330 y Fi(H)1141 4289 y Fl(+)1207 4330 y Fi(;)216 b(\033)1508 4345 y Fl(0)1553 4330 y Fm(\()p Fi(H)1683 4289 y Fe(\000)1750 4330 y Fm(\))27 b(=)h Fi(H)2012 4289 y Fe(\000)2079 4330 y Fi(L)2147 4289 y Fl(+)2213 4330 y Fi(L)2281 4289 y Fe(\000)2348 4330 y Fi(;)216 b(\033)2649 4345 y Fl(0)2693 4330 y Fm(\()p Fi(L)2800 4289 y Fe(\006)2867 4330 y Fm(\))28 b(=)g Fi(H)3130 4289 y Fe(\006)3197 4330 y Fi(H)3288 4289 y Fe(\006)3355 4330 y Fi(:)162 b Fm(\(3)p Fi(:)p Fm(3\))150 4574 y(Since)25 b Fi(L)469 4538 y Fl(+)559 4574 y Fm(and)g Fi(L)812 4538 y Fe(\000)903 4574 y Fm(only)e(app)s(ear)i(in)f(the)g(com)m(bination)f Fi(L)2328 4538 y Fl(+)2394 4574 y Fi(L)2462 4538 y Fe(\000)2529 4574 y Fm(,)j(w)m(e)f(can)f(de\014ne)i(a)e(new)g(sym)m(b)s(ol)150 4693 y Fi(L)31 b Fm(=)h Fi(L)426 4657 y Fl(+)492 4693 y Fi(L)560 4657 y Fe(\000)662 4693 y Fm(and)k(ha)m(v)m(e)g(a)f (substitution)g(system)g(with)g(3)g(elemen)m(ts,)h(whose)g(rule)f(w)m (e)h(denote)150 4813 y Fi(\033)t Fm(:)452 4937 y Fi(\033)t Fm(\()p Fi(H)643 4896 y Fl(+)708 4937 y Fm(\))28 b(=)g Fi(LH)1039 4896 y Fl(+)1105 4937 y Fi(;)215 b(\033)t Fm(\()p Fi(H)1539 4896 y Fe(\000)1606 4937 y Fm(\))27 b(=)h Fi(H)1868 4896 y Fe(\000)1935 4937 y Fi(L;)216 b(\033)t Fm(\()p Fi(L)p Fm(\))26 b(=)i Fi(H)2676 4896 y Fl(+)2742 4937 y Fi(H)2833 4896 y Fl(+)2899 4937 y Fi(H)2990 4896 y Fe(\000)3057 4937 y Fi(H)3148 4896 y Fe(\000)3216 4937 y Fi(:)301 b Fm(\(3)p Fi(:)p Fm(4\))150 5160 y Fn(Lemma)52 b(4:)68 b Fk(The)47 b(se)-5 b(quenc)g(e)46 b Fi(\033)1419 5124 y Fh(n)1472 5160 y Fm(\()p Fi(H)1602 5124 y Fl(+)1668 5160 y Fm(\))h Fk(c)-5 b(ontains)45 b(neither)h(the)h(subse)-5 b(quenc)g(e)45 b Fi(LL)i Fk(nor)f(the)150 5280 y(subse)-5 b(quenc)g(e)35 b Fi(H)786 5244 y Fe(\000)853 5280 y Fi(H)944 5244 y Fl(+)1010 5280 y Fk(.)1900 5519 y Fm(11)p eop %%Page: 12 12 12 11 bop 150 100 a Fk(Pr)-5 b(o)g(of:)47 b Fm(The)36 b(pro)s(of)g(is)g(simple)e(induction.)52 b(The)36 b(only)f(w)m(a)m(y)g (to)h(generate)g(an)f Fi(LL)g Fm(is)h(from)f(an)150 219 y Fi(H)241 183 y Fe(\000)308 219 y Fi(H)399 183 y Fl(+)465 219 y Fm(,)e(and)g(the)g(only)f(w)m(a)m(y)g(to)h(generate)g(an)g Fi(H)2058 183 y Fe(\000)2125 219 y Fi(H)2216 183 y Fl(+)2315 219 y Fm(is)f(from)g Fi(LL)p Fm(.)44 b(Since)33 b(neither)h(app)s(ear) 150 339 y(in)f(the)h(\014rst)f(generation,)g(neither)h(app)s(ears)g(in) f(an)m(y)h(subsequen)m(t)h(generation.)p 3266 349 60 80 v 150 488 a Fn(Lemma)42 b(5:)51 b Fk(The)38 b(se)-5 b(quenc)g(e)38 b Fi(\033)1375 452 y Fh(n)1429 488 y Fm(\()p Fi(H)1559 452 y Fl(+)1624 488 y Fm(\))h Fk(do)-5 b(es)37 b(not)i(c)-5 b(ontain)38 b(a)g(subse)-5 b(quenc)g(e)38 b(of)g(mor)-5 b(e)39 b(than)f(6)150 608 y(c)-5 b(onse)g(cutive)34 b Fi(H)8 b Fk('s.)150 757 y(Pr)-5 b(o)g(of:)42 b Fm(Since)34 b Fi(LL)f Fm(do)s(es)g(not)g(o)s(ccur)h(in)f Fi(\033)1732 721 y Fh(n)p Fe(\000)p Fl(1)1887 757 y Fm(\()p Fi(H)2017 721 y Fl(+)2083 757 y Fm(\),)g(the)g(longest)g(p)s(ossible)g(sequence)i (of)e Fi(H)8 b Fm('s)150 877 y(in)35 b Fi(\033)329 841 y Fh(n)383 877 y Fm(\()p Fi(H)513 841 y Fl(+)578 877 y Fm(\))g(w)m(ould)g(come)g(from)f(a)h(sequence)h Fi(H)2028 841 y Fl(+)2094 877 y Fi(LH)2253 841 y Fe(\000)2355 877 y Fm(in)f Fi(\033)2534 841 y Fh(n)p Fe(\000)p Fl(1)2690 877 y Fm(\()p Fi(H)2820 841 y Fl(+)2886 877 y Fm(\).)48 b(This)35 b(giv)m(es)g(rise)g(to)150 996 y Fi(LH)309 960 y Fl(+)375 996 y Fi(H)466 960 y Fl(+)532 996 y Fi(H)623 960 y Fl(+)689 996 y Fi(H)780 960 y Fe(\000)847 996 y Fi(H)938 960 y Fe(\000)1005 996 y Fi(H)1096 960 y Fe(\000)1163 996 y Fi(L)p Fm(,)e(or)g(6)g Fi(H)8 b Fm('s)33 b(in)g(a)h(ro)m(w.)p 2177 1006 V 349 1146 a(Let)26 b Fi(f)11 b Fm(\()p Fi(n)p Fm(\))24 b(equal)i(the)g(n)m(um)m(b)s(er)g(of)g Fi(L)p Fm('s)f(in)h(the)f(\014rst)h(half)g(of)g(the)g(sequence)h Fi(\033)3178 1110 y Fh(n)3232 1146 y Fm(\()p Fi(H)3362 1110 y Fl(+)3428 1146 y Fm(\))e(min)m(us)150 1265 y(the)j(n)m(um)m(b)s (er)g(of)g Fi(L)p Fm('s)g(in)f(the)h(second)h(half)f(of)g(the)g (sequence.)44 b(As)28 b(w)m(e)g(shall)f(see,)i Fi(f)11 b Fm(\()p Fi(n)p Fm(\))27 b(is)g(closely)150 1385 y(related)33 b(to)f(the)h(exten)m(t)g(to)f(whic)m(h)i(slippage)f(o)s(ccurs)g(along)g (the)g(largest)f(fault)h(line)g(in)f Fi(T)3527 1400 y Fl(2)p Fh(n)p Fl(+2)3722 1385 y Fm(.)150 1534 y Fn(Lemma)38 b(6:)45 b Fk(If)35 b Fj(j)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))p Fj(j)26 b Fi(>)i Fm(6)p Fk(,)35 b(then)h Fj(j)p Fi(f)11 b Fm(\()p Fi(n)21 b Fm(+)h(1\))p Fj(j)27 b(\025)h(j)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))p Fj(j)21 b Fm(+)h(2)p Fk(.)349 1684 y Fm(Let)h Fi(s)g Fm(and)g Fi(s)817 1648 y Fe(0)867 1684 y Fm(denote)h(the)f(\014rst)g(and)g(second)i(halv)m(es) d(of)h Fi(\033)2481 1648 y Fh(n)2535 1684 y Fm(\()p Fi(H)2665 1648 y Fl(+)2731 1684 y Fm(\),)h(resp)s(ectiv)m(ely)-8 b(.)40 b(Supp)s(ose)150 1803 y(that)j Fi(s)g Fm(con)m(tains)h Fi(h)f(H)8 b Fm('s)43 b(and)h Fi(l)h(L)p Fm('s,)g(while)e Fi(s)1972 1767 y Fe(0)2043 1803 y Fm(con)m(tains)h Fi(h)2501 1767 y Fe(0)2572 1803 y Fi(H)8 b Fm('s)43 b(and)h Fi(l)3009 1767 y Fe(0)3080 1803 y Fi(L)p Fm('s.)74 b(Note)42 b(that)150 1923 y Fi(f)11 b Fm(\()p Fi(n)p Fm(\))45 b(=)i Fi(l)32 b Fj(\000)e Fi(l)718 1887 y Fe(0)792 1923 y Fm(=)47 b Fi(h)973 1887 y Fe(0)1031 1923 y Fj(\000)30 b Fi(h)p Fm(.)78 b(Since)46 b(eac)m(h)f Fi(H)52 b Fm(generates)46 b(an)e Fi(H)53 b Fm(and)45 b(an)g Fi(L)p Fm(,)h(while)f(eac)m(h)g Fi(L)150 2042 y Fm(generates)32 b(four)f Fi(H)8 b Fm('s,)31 b Fi(\033)t Fm(\()p Fi(s)p Fm(\))e(con)m(tains)i(4)p Fi(l)19 b Fm(+)f Fi(h)31 b(H)8 b Fm('s)30 b(and)h Fi(h)h(L)p Fm('s,)e(while)h Fi(\033)t Fm(\()p Fi(s)2959 2006 y Fe(0)2986 2042 y Fm(\))f(con)m(tains)h(4)p Fi(l)3525 2006 y Fe(0)3570 2042 y Fm(+)18 b Fi(h)3722 2006 y Fe(0)150 2162 y Fi(H)8 b Fm('s)41 b(and)g Fi(h)607 2126 y Fe(0)676 2162 y Fi(L)p Fm('s.)67 b(Th)m(us)42 b Fi(\033)t Fm(\()p Fi(s)p Fm(\))e(con)m(tains)h (2)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))40 b(more)g(terms,)h(but)h Fi(f)11 b Fm(\()p Fi(n)p Fm(\))39 b(few)m(er)k Fi(L)p Fm('s,)f(than)150 2281 y Fi(\033)t Fm(\()p Fi(s)297 2245 y Fe(0)324 2281 y Fm(\).)349 2431 y(No)m(w)d(supp)s(ose)i Fi(f)11 b Fm(\()p Fi(n)p Fm(\))37 b Fi(>)i Fm(0.)63 b(The)40 b(\014rst)g(half)f(of)h Fi(\033)2261 2395 y Fh(n)p Fl(+1)2416 2431 y Fm(\()p Fi(H)2546 2395 y Fl(+)2612 2431 y Fm(\))f(is)g(all)g(of) h Fi(\033)t Fm(\()p Fi(s)p Fm(\),)f(min)m(us)h(the)150 2550 y(last)34 b Fi(f)11 b Fm(\()p Fi(n)p Fm(\))34 b(elemen)m(ts,)h (while)f(the)h(second)h(half)f(of)g Fi(\033)2140 2514 y Fh(n)p Fl(+1)2295 2550 y Fm(\()p Fi(H)2425 2514 y Fl(+)2491 2550 y Fm(\))f(is)h(the)g(last)f Fi(f)11 b Fm(\()p Fi(n)p Fm(\))34 b(elemen)m(ts)h(of)150 2670 y Fi(\033)t Fm(\()p Fi(s)p Fm(\))g(and)i(all)f(of)h Fi(\033)t Fm(\()p Fi(s)974 2634 y Fe(0)1001 2670 y Fm(\).)53 b(Th)m(us)38 b Fi(f)11 b Fm(\()p Fi(n)23 b Fm(+)i(1\))36 b(equals)h Fj(\000)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))35 b(min)m(us)i(t)m(wice)f(the)h(n)m(um) m(b)s(er)g(of)g Fi(L)p Fm('s)150 2790 y(in)f(the)g(last)f Fi(f)11 b Fm(\()p Fi(n)p Fm(\))35 b(elemen)m(ts)h(of)g Fi(\033)t Fm(\()p Fi(s)p Fm(\).)51 b(Since)36 b Fi(f)11 b Fm(\()p Fi(n)p Fm(\))31 b Fi(>)i Fm(6,)j(there)g(m)m(ust)g(b)s(e)g (at)f(least)h(one)g Fi(L)g Fm(in)150 2909 y(the)e(last)e Fi(f)11 b Fm(\()p Fi(n)p Fm(\))32 b(elemen)m(ts)i(of)f Fi(\033)t Fm(\()p Fi(s)p Fm(\),)f(so)h Fj(j)p Fi(f)11 b Fm(\()p Fi(n)21 b Fm(+)i(1\))p Fj(j)j Fm(=)j Fj(\000)p Fi(f)11 b Fm(\()p Fi(n)21 b Fm(+)i(1\))k Fj(\025)h Fi(f)11 b Fm(\()p Fi(n)p Fm(\))21 b(+)i(2.)349 3059 y(If)35 b Fi(f)11 b Fm(\()p Fi(n)p Fm(\))29 b Fi(<)h Fm(0,)35 b(then)g(the)g (\014rst)g(half)g(of)g Fi(\033)1879 3022 y Fh(n)p Fl(+1)2033 3059 y Fm(\()p Fi(H)2163 3022 y Fl(+)2229 3059 y Fm(\))f(is)h(all)f(of) g Fi(\033)t Fm(\()p Fi(s)p Fm(\),)g(plus)h(the)g(\014rst)g Fj(j)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))p Fj(j)150 3178 y Fm(elemen)m(ts)38 b(of)f Fi(\033)t Fm(\()p Fi(s)826 3142 y Fe(0)853 3178 y Fm(\).)56 b(W)-8 b(e)38 b(then)g(ha)m(v)m(e)g Fi(f)11 b Fm(\()p Fi(n)24 b Fm(+)i(1\))37 b(equalling)f Fj(\000)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))37 b(plus)h(t)m(wice)g(the)f (n)m(um)m(b)s(er)150 3298 y(of)c Fi(L)p Fm('s)g(in)g(the)h(\014rst)f Fj(j)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))p Fj(j)32 b Fm(elemen)m(ts)h(of)g Fi(\033)t Fm(\()p Fi(s)p Fm(\).)43 b(By)32 b(Lemma)f(5,)i(there)h(m)m (ust)f(b)s(e)g(at)g(least)g(one)150 3417 y(suc)m(h)i Fi(L)p Fm(,)d(so)i Fj(j)p Fi(f)11 b Fm(\()p Fi(n)21 b Fm(+)h(1\))27 b(=)i Fi(f)11 b Fm(\()p Fi(n)21 b Fm(+)h(1\))28 b Fj(\025)g Fm(2)22 b Fj(\000)g Fi(f)11 b Fm(\()p Fi(n)p Fm(\))27 b(=)h Fj(j)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))p Fj(j)20 b Fm(+)j(2.)p 2712 3427 V 150 3567 a Fn(Lemma)38 b(7:)45 b Fm(lim)828 3582 y Fh(n)p Fe(!1)1058 3567 y Fj(j)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))p Fj(j)25 b Fm(=)k(+)p Fj(1)p Fm(.)150 3716 y Fk(Pr)-5 b(o)g(of:)73 b Fm(By)47 b(explicit)g(computation,)k Fi(f)11 b Fm(\(1\))51 b(=)i(1,)f Fi(f)11 b Fm(\(2\))51 b(=)i Fj(\000)p Fm(1,)f Fi(f)11 b Fm(\(3\))51 b(=)i(1,)f Fi(f)11 b Fm(\(4\))51 b(=)i Fj(\000)p Fm(3,)150 3836 y Fi(f)11 b Fm(\(5\))27 b(=)i(3,)34 b Fi(f)11 b Fm(\(6\))27 b(=)i Fj(\000)p Fm(5)34 b(and)h Fi(f)11 b Fm(\(7\))27 b(=)i(9.)46 b(By)33 b(Lemma)f(6,)h(for)h Fi(n)29 b Fj(\025)g Fm(7,)k Fj(j)p Fi(f)11 b Fm(\()p Fi(n)21 b Fm(+)i(1\))p Fj(j)28 b Fi(>)h Fj(j)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))p Fj(j)p Fm(,)32 b(so)150 3955 y Fj(j)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))p Fj(j)26 b(\025)i Fi(n)22 b Fm(+)h(2)33 b(go)s(es)g(to)f(in\014nit)m(y)i(as)f Fi(n)27 b Fj(!)h(1)p Fm(.)p 2052 3965 V 349 4105 a(W)-8 b(e)29 b(ha)m(v)m(e)g(pro)m(v)m(en)h(that)e Fj(j)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))p Fj(j)27 b Fm(gro)m(ws)h(without)h(b)s(ound,)h (whic)m(h)f(is)g(all)f(that)g(w)m(e)h(need.)44 b(In)150 4224 y(fact,)34 b Fj(j)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))p Fj(j)32 b Fm(gro)m(ws)i(exp)s(onen)m(tially)-8 b(.)44 b(F)-8 b(or)34 b(large)f Fi(n)p Fm(,)g(an)h(appro)m(ximate)f(fraction)h (2)p Fi(=)p Fm(\(3)22 b(+)3529 4142 y Fj(p)p 3612 4142 100 4 v 82 x Fm(17)o(\))150 4344 y(of)35 b(the)g(elemen)m(ts)f(of)h Fi(\033)1022 4308 y Fh(n)1076 4344 y Fm(\()p Fi(H)1206 4308 y Fl(+)1272 4344 y Fm(\))f(are)h Fi(L)p Fm('s,)f(so)h Fi(f)11 b Fm(\()p Fi(n)22 b Fm(+)h(1\))30 b Fj(\031)g(\000)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))23 b(+)g(2)p Fi(f)11 b Fm(\()p Fi(n)p Fm(\))22 b Fj(\002)h Fm(2)p Fi(=)p Fm(\(3)g(+)3421 4261 y Fj(p)p 3504 4261 V 83 x Fm(17\))30 b(=)150 4463 y Fi(f)11 b Fm(\()p Fi(n)p Fm(\)\(1)21 b Fj(\000)558 4381 y(p)p 641 4381 V 82 x Fm(17\))p Fi(=)p Fm(2.)45 b(The)35 b(gro)m(wth)f(rate,)f(\(1)23 b Fj(\000)1940 4381 y(p)p 2023 4381 V 82 x Fm(17)o(\))p Fi(=)p Fm(2,)33 b(equals)h Fi(\025)2683 4478 y Fe(\000)2751 4463 y Fm(,)g(the)g(second) h(eigen)m(v)-6 b(alue)150 4583 y(of)33 b(the)h(p)s(opulation)f(matrix)e Fi(M)11 b Fm(.)349 4732 y(W)-8 b(e)30 b(no)m(w)f(return)h(to)f(the)g (question)g(of)h(slippage)f(along)g(fault)g(lines.)43 b(Consider)29 b(t)m(w)m(o)g(large)150 4852 y(triangles)36 b(that)h(meet)f(h)m(yp)s(oten)m(use)i(to)f(h)m(yp)s(oten)m(use)h(to)f (form)f(a)g(rectangle,)i(as)f(in)g(Figure)g(7.)150 4971 y(Let)g Fi(P)51 b Fm(and)38 b Fi(R)f Fm(b)s(e)h(the)f(ends)i(of)e(the)g (common)f(h)m(yp)s(oten)m(use,)j(and)f(let)f Fi(Q)f Fm(b)s(e)i(the)f 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b(limiting)c(distribution)j(is)f(giv)m(en)h(b)m(y)g(the)g(eigen)m(v)m (ector)g(asso)s(ciated)150 2521 y(to)35 b(the)i(largest)e(eigen)m(v)-6 b(alue)37 b(of)f(a)g Fi(p)24 b Fj(\002)g Fi(p)36 b Fm(\(or)g Fi(q)28 b Fj(\002)c Fi(q)t Fm(\))36 b(p)s(opulation)g(matrix.)49 b(W)-8 b(e)37 b(also)e(analyze)150 2640 y(the)j(second)h(eigen)m(v)-6 b(alue)38 b(of)g(this)g(matrix.)54 b(The)39 b(failure)e(of)h(the)g (edge-to-edge)h(prop)s(ert)m(y)f(for)150 2760 y Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(2\))33 b(w)m(as)g(a)h(result)g(of)g (\015uctuations)g(that)f(w)m(ere)i(go)m(v)m(erned)f(b)m(y)g(this)g (second)h(eigen)m(v)-6 b(alue.)150 2879 y(W)e(e)23 b(conjecture)h(that) e(the)h(edge-to-edge)h(prop)s(ert)m(y)f(holds)g(only)f(for)h(those)g (tilings)e(with)i(second)150 2999 y(eigen)m(v)-6 b(alue)34 b(smaller)e(than)h(1,)g(and)h(w)m(e)g(classify)f(these)h(tilings.)349 3148 y(W)-8 b(e)33 b(then)h(turn)f(to)g(the)g(join)m(t)g(distribution)g (of)g(sizes)g(and)h(orien)m(tations)e(in)h(rational)f Fi(p=q)150 3268 y Fm(tilings.)40 b(W)-8 b(e)24 b(sho)m(w)g(that,)h(for) f(eac)m(h)h(size,)g(the)f(distribution)g(of)g(orien)m(tations)f(is)h (asymptotically)150 3387 y(uniform.)42 b(Sp)s(eci\014cally)-8 b(,)29 b(w)m(e)h(parametrize)e Fi(O)s Fm(\(2\),)g(the)h(group)g(of)g (orien)m(tations,)h(b)m(y)f(t)m(w)m(o)f(copies)150 3507 y(of)44 b(the)g(unit)g(circle.)76 b(Giv)m(en)45 b(an)f(in)m(terv)-6 b(al)43 b(in)h(this)g(set,)j(the)d(fraction)g(of)g(tiles)g(in)f Fi(T)3447 3522 y Fh(n)3502 3507 y Fm(,)j(of)e(a)150 3626 y(giv)m(en)35 b(size,)i(whose)f(orien)m(tations)f(lie)g(in)h(that)f(in) m(terv)-6 b(al,)36 b(approac)m(hes)h(a)e(constan)m(t)h(times)f(the)150 3746 y(length)f(of)h(the)f(in)m(terv)-6 b(al.)47 b(In)34 b(the)h(terminology)c(of)k(Radin)f([R3],)g(the)g(tilings)g Fi(T)14 b Fm(\()p Fi(p=q)t Fm(\))33 b(exhibit)150 3866 y(\\statistical)f(rotational)f(symmetry".)349 4015 y(By)46 b(Theorem)g(3,)j Fi(T)14 b(il)r Fm(\()p Fi(p=q)t Fm(\))46 b(con)m(tains)h(triangles)f(of)g Fi(m)k Fm(=)g(max)o(\()p Fi(p;)17 b(q)t Fm(\))45 b(distinct)h(sizes.)150 4134 y Fi(T)14 b(il)r Fm(\()p Fi(p=q)t Fm(\))28 b(is)h(equiv)-6 b(alen)m(t)29 b(to)f(a)h(traditional)e(substitution)j(tiling,)e(with)g (prototiles)h Fi(D)3370 4149 y Fl(1)3415 4134 y Fi(;)17 b(:)g(:)g(:)d(D)3674 4149 y Fh(m)150 4254 y Fm(of)25 b Fi(m)h Fm(sizes.)42 b(W)-8 b(e)25 b(tak)m(e)g Fi(D)1085 4269 y Fl(1)1155 4254 y Fm(to)g(b)s(e)g(the)g(largest)g(size)h(and)f Fi(D)2321 4269 y Fh(m)2422 4254 y Fm(to)g(b)s(e)g(the)h(smallest.)40 b(Sub)s(division)150 4374 y(and)34 b(linear)f(rescaling)g(b)m(y)h Fi(r)1218 4337 y Fe(\000)p Fl(1)1325 4374 y Fm(,)f(where)h Fi(r)c Fm(=)f(\()p Fi(a=c)p Fm(\))2079 4337 y Fl(1)p Fh(=p)2232 4374 y Fm(=)f(\()p Fi(b=)p Fm(2)p Fi(c)p Fm(\))2601 4337 y Fl(1)p Fh(=q)2725 4374 y Fm(,)33 b(tak)m(es)g Fi(D)3123 4389 y Fh(i)p Fl(+1)3291 4374 y Fm(to)g Fi(D)3495 4389 y Fh(i)3528 4374 y Fm(,)g(and)150 4493 y(tak)m(es)g Fi(D)487 4508 y Fl(1)565 4493 y Fm(to)g(four)g(copies)h(of)f Fi(D)1383 4508 y Fh(q)1461 4493 y Fm(and)g(one)h(cop)m(y)f(of)g Fi(D)2264 4508 y Fh(p)2311 4493 y Fm(.)44 b(That)33 b(is,)f(the)i(p)s (opulation)e(matrix,)150 4613 y(whic)m(h)41 b(giv)m(es)g(the)f(p)s (opulation)h(of)f Fi(T)1559 4628 y Fh(n)p Fl(+1)1755 4613 y Fm(in)h(terms)e(of)i(the)g(p)s(opulation)f(of)h Fi(T)3150 4628 y Fh(n)3204 4613 y Fm(,)h(has)f(matrix)150 4732 y(elemen)m(ts)1254 5131 y Fi(M)1351 5146 y Fh(ij)1448 5131 y Fm(=)1554 4897 y Fc(8)1554 4986 y(>)1554 5016 y(<)1554 5196 y(>)1554 5225 y(:)1659 4961 y Fm(1)99 b(if)34 b Fi(j)f Fm(=)28 b Fi(i)23 b Fm(+)f(1;)1659 5081 y(1)99 b(if)34 b Fi(j)f Fm(=)28 b(1)33 b(and)h Fi(i)28 b Fm(=)g Fi(p)p Fm(;)1659 5200 y(4)99 b(if)34 b Fi(j)f Fm(=)28 b(1)33 b(and)h Fi(i)28 b Fm(=)g Fi(q)t Fm(;)1659 5320 y(0)99 b(otherwise.)3545 5131 y(\(4)p Fi(:)p Fm(1\))1900 5519 y(14)p eop %%Page: 15 15 15 14 bop 349 100 a Fm(The)34 b(prop)s(erties)g(of)f Fi(M)44 b Fm(are)33 b(summarized)g(in)g(the)g(follo)m(wing)g(theorem:) 150 249 y Fn(Theorem)38 b(6)207 399 y Fk(1\))50 b(The)36 b(char)-5 b(acteristic)34 b(p)-5 b(olynomial)36 b(of)f Fi(M)46 b Fk(is)1264 685 y Fi(p)p Fm(\()p Fi(\025)p Fm(\))28 b(=)1583 544 y Fc(\032)1674 630 y Fi(\025)1732 594 y Fh(q)1799 630 y Fj(\000)22 b Fi(\025)1956 594 y Fh(q)s Fe(\000)p Fh(p)2126 630 y Fj(\000)g Fm(4)154 b Fk(if)35 b Fi(p)27 b(<)i(q)t Fk(;)1674 750 y Fi(\025)1732 713 y Fh(p)1801 750 y Fj(\000)22 b Fm(4)p Fi(\025)2008 713 y Fh(p)p Fe(\000)p Fh(q)2178 750 y Fj(\000)g Fm(1)102 b Fk(if)35 b Fi(p)27 b(>)i(q)t Fk(.)3545 685 y Fm(\(4)p Fi(:)p Fm(2\))207 1007 y Fk(2\))50 b(The)36 b(lar)-5 b(gest)35 b(eigenvalue)h(of)f Fi(M)46 b Fk(is)35 b Fi(r)1764 971 y Fe(\000)p Fl(2)1870 1007 y Fk(.)207 1157 y(3\))50 b(Ther)-5 b(e)36 b(ar)-5 b(e)35 b(exactly)h Fi(q)k Fk(eigenvalues)35 b(with)g(mo)-5 b(dulus)37 b(gr)-5 b(e)g(ater)35 b(than)h(one.)207 1306 y(4\))50 b(The)36 b(eigenve)-5 b(ctors)34 b Fi( )39 b Fk(of)c Fi(M)11 b Fk(,)35 b(for)g(\014xe)-5 b(d)35 b(eigenvalue)h Fi(\025)p Fk(,)f(take)g(the)h(form)937 1543 y Fi( )1002 1558 y Fh(k)1078 1543 y Fm(=)29 b Fi(\025)1242 1502 y Fh(k)1313 1543 y Fj(\000)23 b Fi(\025)1471 1502 y Fh(k)r Fe(\000)p Fh(p)1623 1543 y Fi(H)8 b Fm(\()p Fi(k)25 b Fj(\000)d Fi(p)h Fj(\000)f Fm(1\))g Fj(\000)h Fm(4)p Fi(\025)2420 1502 y Fh(k)r Fe(\000)p Fh(q)2570 1543 y Fi(H)8 b Fm(\()p Fi(k)25 b Fj(\000)d Fi(q)k Fj(\000)d Fm(1\))p Fi(;)382 b Fm(\(4)p Fi(:)p Fm(3\))349 1779 y Fk(wher)-5 b(e)36 b Fi(H)8 b Fm(\()p Fi(n)p Fm(\))34 b Fk(is)h(the)h(discr)-5 b(ete)34 b(He)-5 b(avyside)35 b(function)1527 2040 y Fi(H)8 b Fm(\()p Fi(n)p Fm(\))27 b(=)1889 1930 y Fc(n)1972 1990 y Fm(1)101 b Fk(if)35 b Fi(n)27 b Fj(\025)i Fm(0)p Fk(;)1972 2110 y Fm(0)101 b Fk(otherwise.)3545 2040 y Fm(\(4)p Fi(:)p Fm(4\))207 2331 y Fk(5\))50 b(Asymptotic)-5 b(al)5 b(ly,)37 b(the)e(numb)-5 b(er)36 b(of)f(tiles)h(of)f(size)f Fi(D)2296 2346 y Fh(k)2382 2331 y Fk(is)g(a)i(fr)-5 b(action)1105 2622 y Fi(\027)1154 2637 y Fh(k)1230 2622 y Fm(=)1348 2554 y(1)21 b Fj(\000)i Fi(r)1567 2518 y Fl(2)p 1348 2599 264 4 v 1411 2690 a Fm(4)p Fi(c)1504 2661 y Fl(2)1623 2622 y Fm(\()p Fi(a)1715 2580 y Fl(2)1760 2622 y Fi(H)8 b Fm(\()p Fi(p)21 b Fj(\000)i Fi(k)s Fm(\))f(+)h Fi(b)2320 2580 y Fl(2)2364 2622 y Fi(H)8 b Fm(\()p Fi(q)26 b Fj(\000)d Fi(k)s Fm(\)\))p Fi(r)2845 2580 y Fe(\000)p Fl(2)p Fh(k)3545 2622 y Fm(\(4)p Fi(:)p Fm(5\))349 2876 y Fk(of)35 b(the)h(total.)207 3025 y(6\))50 b(Asymptotic)-5 b(al)5 b(ly,)37 b(the)e(ar)-5 b(e)g(a)35 b(c)-5 b(over)g(e)g(d)35 b(by)h(tiles)f(of)g(size)g Fi(D)2517 3040 y Fh(k)2602 3025 y Fk(is)g(a)g(fr)-5 b(action)1373 3316 y Fi(\032)1425 3331 y Fh(k)1502 3316 y Fm(=)1619 3248 y Fi(a)1672 3212 y Fl(2)1716 3248 y Fi(H)8 b Fm(\()p Fi(p)22 b Fj(\000)h Fi(k)s Fm(\))f(+)g Fi(b)2276 3212 y Fl(2)2321 3248 y Fi(H)8 b Fm(\()p Fi(q)25 b Fj(\000)e Fi(k)s Fm(\))p 1619 3293 1096 4 v 1964 3384 a Fi(pa)2067 3355 y Fl(2)2134 3384 y Fm(+)f Fi(q)t(b)2324 3355 y Fl(2)3545 3316 y Fm(\(4)p Fi(:)p Fm(6\))150 3590 y Fk(of)35 b(the)h(total.)349 3739 y Fm(Note)45 b(that,)j(if)d Fi(p)j(<)h(q)t Fm(,)f(then)e Fi(H)8 b Fm(\()p Fi(k)33 b Fj(\000)e Fi(q)j Fj(\000)d Fm(1\))45 b(is)g(iden)m(tically)g(zero)g(and)h Fi(H)8 b Fm(\()p Fi(q)34 b Fj(\000)d Fi(k)s Fm(\))45 b(is)150 3859 y(iden)m(tically)d(one.)73 b(If)43 b Fi(p)h(>)g(q)t Fm(,)h(then)f Fi(H)8 b Fm(\()p Fi(k)31 b Fj(\000)e Fi(p)g Fj(\000)g Fm(1\))42 b(is)h(iden)m(tically)f(zero)h(and)g Fi(H)8 b Fm(\()p Fi(p)28 b Fj(\000)h Fi(k)s Fm(\))43 b(is)150 3978 y(iden)m(tically)g(one.)75 b(As)44 b(written,)h (expressions)g(\(4.3\),)f(\(4.5\))f(and)h(\(4.6\))e(apply)h(to)g(b)s (oth)h(the)150 4098 y Fi(p)28 b(<)g(q)37 b Fm(and)d Fi(p)28 b(>)g(q)37 b Fm(cases.)150 4247 y Fn(Corollary:)50 b Fk(L)-5 b(et)39 b Fi(t)f Fk(b)-5 b(e)38 b(a)g(tile)g(of)g(size)f Fi(k)s Fk(.)54 b(L)-5 b(et)38 b Fi(N)2049 4262 y Fh(k)2098 4247 y Fm(\()p Fi(n)p Fm(\))g Fk(b)-5 b(e)38 b(the)g(numb)-5 b(er)39 b(of)e(desc)-5 b(endants)37 b(of)h Fi(t)150 4367 y Fk(after)d(the)h Fi(n)p Fk(-th)f(applic)-5 b(ation)34 b(of)h(the)h(substitution)f(rule.)46 b(Then)1164 4655 y Fm(lim)1130 4715 y Fh(n)p Fe(!1)1354 4655 y Fi(r)1402 4614 y Fl(2)p Fh(n)1496 4655 y Fi(N)1576 4670 y Fh(k)1625 4655 y Fm(\()p Fi(n)p Fm(\))27 b(=)2182 4588 y(4)p Fi(c)2275 4551 y Fl(2)2320 4588 y Fi(r)2368 4551 y Fl(2)p Fh(k)p 1907 4632 824 4 v 1907 4723 a Fm(\(1)22 b Fj(\000)g Fi(r)2165 4695 y Fl(2)2210 4723 y Fm(\)\()p Fi(pa)2391 4695 y Fl(2)2457 4723 y Fm(+)h Fi(q)t(b)2648 4695 y Fl(2)2692 4723 y Fm(\))2743 4655 y Fi(:)774 b Fm(\(4)p Fi(:)p Fm(7\))150 5041 y Fk(Pr)-5 b(o)g(of)38 b(of)h(Cor)-5 b(ol)5 b(lary:)54 b Fm(F)-8 b(rom)35 b(the)j(distribution)f(\(4.5\),)g(w)m(e)h(compute)f(the)h(a)m (v)m(erage)f(area)h(p)s(er)150 5160 y(tile)g(to)f(b)s(e)i(\(1)25 b Fj(\000)h Fi(r)859 5124 y Fl(2)903 5160 y Fm(\))p Fi(ab)p Fm(\()p Fi(pa)1180 5124 y Fl(2)1249 5160 y Fm(+)g Fi(q)t(b)1443 5124 y Fl(2)1487 5160 y Fm(\))p Fi(=)p Fm(8)p Fi(c)1669 5124 y Fl(2)1713 5160 y Fm(.)59 b(A)38 b(tile)g(of)g(size)h Fi(D)2484 5175 y Fh(k)2533 5160 y Fm(,)g(sub)s(divided)g(and)g (rescaled)g Fi(n)150 5280 y Fm(times,)32 b(has)i(area)f Fi(abr)982 5244 y Fl(2)p Fh(k)r Fe(\000)p Fl(2)p Fh(n)1221 5280 y Fi(=)p Fm(2.)43 b(Dividing)33 b(b)m(y)g(the)h(area)f(p)s(er)g (unit)h(tile)e(w)m(e)i(obtain)f(\(4.7\).)p 3674 5290 60 80 v 1900 5519 a(15)p eop %%Page: 16 16 16 15 bop 150 100 a Fn(Remark:)57 b Fm(If)41 b Fi(q)h(>)d Fm(1,)i(then)g(the)f(second)h(largest)f(eigen)m(v)-6 b(alue)40 b(of)g Fi(M)50 b Fm(is)40 b(greater)g(than)g(one.)150 219 y(The)47 b(\015uctuations)f(in)h(p)s(opulation)e(asso)s(ciated)h (to)g(the)g(corresp)s(onding)i(eigen)m(v)m(ector)e(then)150 339 y(gro)m(w)41 b(with)f(sub)s(division,)i(although)f(they)f(do)h(not) g(gro)m(w)f(as)h(fast)g(as)g(the)g(p)s(opulation)f(as)h(a)150 458 y(whole.)48 b(It)34 b(w)m(as)h(precisely)f(this)h(phenomenon)g (that)f(caused)i(the)f(edge-to-edge)h(prop)s(ert)m(y)e(to)150 578 y(fail)40 b(for)g Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(2\).)63 b(If)40 b Fi(q)j Fm(=)c(1,)i(then)g(the)f(second)i(eigen)m (v)-6 b(alue)40 b(is)g(less)h(than)f(one.)64 b(In)41 b Fj(x)p 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b(the)i(char)-5 b(acteristic)34 b(function)h(of)g (an)g(interval)h(in)f Fi(X)8 b Fk(.)150 4682 y(Pr)-5 b(o)g(of:)64 b Fm(W)-8 b(e)45 b(\014rst)f(reduce)i(the)e(problem)g(to)f (establishing)h(\(4.10a\))f(for)h(an)g(arbitrary)f(con-)150 4802 y(tin)m(uous)37 b(function)g Fi(f)11 b Fm(.)52 b(Once)37 b(w)m(e)g(ha)m(v)m(e)f(established)h(\(4.10a\),)e(\(4.10b\))g(follo)m (ws)h(b)m(y)g(applying)150 4921 y(\(4.10a\))g(to)i(the)g(con)m(tin)m (uous)h(function)1715 4895 y(~)1693 4921 y Fi(f)11 b Fm(\()p Fi(k)s(;)17 b Fm(\012\))34 b(=)i Fi(r)2198 4885 y Fl(2)p Fh(k)2286 4921 y Fi(f)11 b Fm(\()p Fi(k)s(;)17 b Fm(\012\).)56 b(Once)39 b(equations)f(\(4.10ab\))150 5041 y(ha)m(v)m(e)29 b(b)s(een)g(established)g(for)g(con)m(tin)m(uous)g (functions,)i(the)d(extension)h(to)f(c)m(haracteristic)g(func-)150 5160 y(tions)h(is)g(standard)g(\(for)g(details)g(see)g([CFS]\).)f(Let)i Fi(I)36 b Fm(b)s(e)29 b(an)g(in)m(terv)-6 b(al,)29 b(and)h(let)e Fi(\037)3190 5175 y Fh(I)3265 5160 y Fm(b)s(e)i(its)e(c)m(har-)150 5280 y(acteristic)36 b(function.)52 b(W)-8 b(e)36 b(c)m(ho)s(ose)h(con) m(tin)m(uous)g(functions)g Fi(f)2502 5244 y Fe(\006)2491 5305 y Fh(\017)2569 5280 y Fm(,)f(suc)m(h)i(that)d Fi(f)3139 5244 y Fe(\000)3128 5305 y Fh(\017)3238 5280 y Fj(\024)d Fi(\037)3409 5295 y Fh(I)3487 5280 y Fj(\024)h Fi(f)3657 5244 y Fl(+)3646 5305 y Fh(\017)3722 5280 y Fm(,)1900 5519 y(17)p eop %%Page: 18 18 18 17 bop 150 100 a Fm(and)45 b(suc)m(h)i(that)d(lim)958 115 y Fh(\017)p Fe(!)p Fl(0)1132 19 y Fc(R)1179 135 y Fh(X)1271 100 y Fi(f)1331 63 y Fe(\006)1320 124 y Fh(\017)1398 100 y Fi(d\027)53 b Fm(=)1677 19 y Fc(R)1724 135 y Fh(I)1787 100 y Fi(d\027)6 b Fm(.)79 b(lim)2139 115 y Fh(n)p Fe(!1)2400 100 y Fi(<)47 b(\037)2586 115 y Fh(I)2632 100 y Fi(;)17 b(S)2738 115 y Fh(n)2839 100 y Fi(>)46 b Fm(is)f(sandwic)m(hed)h(b)s (e-)150 219 y(t)m(w)m(een)37 b(lim)574 234 y Fh(n)p Fe(!1)819 219 y Fi(<)c(f)989 183 y Fe(\000)978 244 y Fh(\017)1056 219 y Fi(;)17 b(S)1162 234 y Fh(n)1248 219 y Fi(>)37 b Fm(and)g(lim)1697 234 y Fh(n)p Fe(!1)1943 219 y Fi(<)c(f)2113 183 y Fl(+)2102 244 y Fh(\017)2178 219 y Fi(;)17 b(S)2284 234 y Fh(n)2370 219 y Fi(>)p Fm(,)37 b(hence)h(b)s(et)m(w)m(een)3180 139 y Fc(R)3227 255 y Fh(X)3319 219 y Fi(f)3379 183 y Fe(\000)3368 244 y Fh(\017)3446 219 y Fi(d\027)k Fm(and)150 258 y Fc(R)197 374 y Fh(X)289 339 y Fi(f)349 303 y Fl(+)338 363 y Fh(\017)415 339 y Fi(d\027)6 b Fm(,)33 b(and)h(so)f(m)m(ust)g (equal)1409 258 y Fc(R)1456 374 y Fh(I)1518 339 y Fi(d\027)6 b Fm(.)349 488 y(T)-8 b(o)32 b(establish)h(\(4.10a\))d(w)m(e)j(m)m(ust) f(in)m(tro)s(duce)h(some)f(notation.)43 b(If)33 b Fi(f)43 b Fm(is)32 b(a)g(function)h(on)g Fi(X)150 608 y Fm(and)g Fi(S)39 b Fm(is)33 b(a)g(collection)f(of)h(tiles,)f(let)h(\()p Fi(f)5 b(;)17 b(S)6 b Fm(\))31 b(b)s(e)j(the)f(sum)g(of)g Fi(f)43 b Fm(ev)-6 b(aluated)33 b(on)g(the)h(individual)150 727 y(tiles)g(of)g Fi(S)6 b Fm(.)47 b(Let)35 b(\010)f(denote)h(the)g (action)f(of)g(sub)s(dividing)h(and)g(rescaling.)47 b(That)34 b(is,)g(\010)g(acting)150 847 y(on)e(a)f(tile)h(of)g(size)g Fi(D)920 862 y Fh(k)r Fl(+1)1102 847 y Fm(giv)m(es)f(a)h(tile)f(of)h (size)g Fi(D)1977 862 y Fh(k)2059 847 y Fm(of)g(the)g(same)f(orien)m (tation,)g(while)h(\010)f(acting)150 966 y(on)42 b(a)g(tile)g(of)g (size)h Fi(D)972 981 y Fl(1)1059 966 y Fm(giv)m(es)f(one)g(tile)g(of)g (size)h Fi(D)2086 981 y Fh(p)2174 966 y Fm(and)g(four)g(of)f(size)h Fi(D)2996 981 y Fh(q)3040 966 y Fm(,)h(ha)m(ving)e(v)-6 b(arious)150 1086 y(orien)m(tations.)44 b(Let)33 b(\010)985 1050 y Fe(\003)1064 1086 y Fm(b)s(e)h(the)f(dual)h(of)f(\010)g(b)m(y)h (\()p Fj(\001)p Fi(;)17 b Fj(\001)p Fm(\),)31 b(acting)i(on)g(the)h (space)g(of)f(functions:)1494 1295 y(\(\010)1605 1254 y Fe(\003)1650 1295 y Fi(f)5 b(;)17 b(S)6 b Fm(\))27 b(=)h(\()p Fi(f)5 b(;)17 b Fm(\010\()p Fi(S)6 b Fm(\)\))p Fi(:)1086 b Fm(\(4)p Fi(:)p Fm(11\))150 1504 y(Let)35 b Fi(f)379 1519 y Fh(n)463 1504 y Fm(=)c(\(\010)682 1468 y Fe(\003)728 1504 y Fm(\))767 1468 y Fh(n)821 1504 y Fi(f)11 b Fm(.)48 b(Note)34 b(that)h(\010)1489 1468 y Fe(\003)1569 1504 y Fm(is)g(linear)g(and)g(sends)i(non-negativ)m(e)e (functions)h(to)e(non-)150 1623 y(negativ)m(e)f(functions,)h(so)f(if)h Fi(f)k Fj(\024)28 b Fi(g)t Fm(,)k(then)i Fi(f)1794 1638 y Fh(n)1876 1623 y Fj(\024)28 b Fi(g)2029 1638 y Fh(n)2083 1623 y Fm(.)349 1773 y(No)m(w)g(supp)s(ose)h(w)m(e)f(ha)m(v)m(e)h(a)e (sequence)j(of)e(tiles)g Fi(t)2141 1788 y Fh(n)2223 1773 y Fm(and)g(sup)s(ertiles)h Fi(S)2914 1788 y Fh(n)2996 1773 y Fm(=)f(\010)3173 1737 y Fh(n)3227 1773 y Fi(t)3263 1788 y Fh(n)3317 1773 y Fm(.)42 b(W)-8 b(e)29 b(ha)m(v)m(e)150 1892 y(that)909 2035 y Fi(<)f(f)5 b(;)17 b(S)1174 2050 y Fh(n)1256 2035 y Fi(>)p Fm(=)1641 1967 y(\()p Fi(f)5 b(;)17 b(S)1840 1982 y Fh(n)1893 1967 y Fm(\))p 1450 2012 673 4 v 1450 2103 a(#)33 b(of)h(tiles)f(in)g Fi(S)2068 2118 y Fh(n)2162 2035 y Fm(=)2458 1967 y(\()p Fi(f)2546 1982 y Fh(n)2599 1967 y Fi(;)17 b(t)2680 1982 y Fh(n)2734 1967 y Fm(\))p 2279 2012 V 2279 2103 a(#)33 b(of)h(tiles)f(in)g Fi(S)2897 2118 y Fh(n)2963 2035 y Fi(:)504 b Fm(\(4)p Fi(:)p Fm(12\))150 2253 y(Since)38 b(the)f(n)m(um)m(b)s(er)g(of)g (tiles)g(is)f(giv)m(en)h(asymptotically)d(b)m(y)j(\(4.7\),)f(\(4.10a\)) f(is)i(equiv)-6 b(alen)m(t)36 b(to)150 2373 y Fi(r)198 2337 y Fl(2)p Fh(n)292 2373 y Fi(f)341 2388 y Fh(n)428 2373 y Fm(con)m(v)m(erging)e(uniformly)e(to)1499 2293 y Fc(R)1546 2408 y Fh(X)1639 2373 y Fi(f)11 b(d\027)39 b Fm(times)1277 2650 y Fi(\020)1321 2665 y Fl(0)1366 2650 y Fm(\()p Fi(k)s(;)17 b Fm(\012\))26 b(=)2034 2582 y(4)p Fi(c)2127 2546 y Fl(2)2172 2582 y Fi(r)2220 2546 y Fl(2)p Fh(k)p 1759 2627 824 4 v 1759 2718 a Fm(\(1)c Fj(\000)h Fi(r)2018 2689 y Fl(2)2062 2718 y Fm(\)\()p Fi(pa)2243 2689 y Fl(2)2309 2718 y Fm(+)g Fi(q)t(b)2500 2689 y Fl(2)2544 2718 y Fm(\))2595 2650 y Fi(:)872 b Fm(\(4)p Fi(:)p Fm(13\))349 2937 y(W)-8 b(e)24 b(examine)f(the)h(sp)s (ectrum)g(of)g(the)g(linear)g(op)s(erator)f(\010)2478 2901 y Fe(\003)2548 2937 y Fm(on)h(the)g(function)g(space)h Fi(C)7 b Fm(\()p Fi(X)h Fm(\).)150 3057 y(The)34 b(k)m(ey)f(lemma,)d (whose)k(pro)s(of)g(w)m(e)f(defer,)i(is)150 3206 y Fn(Lemma)48 b(8)41 b Fk(The)i(sp)-5 b(e)g(ctrum)44 b(of)e Fm(\010)1498 3170 y Fe(\003)1587 3206 y Fk(is)h(pur)-5 b(e)43 b(p)-5 b(oint.)67 b Fi(\020)2291 3221 y Fl(0)2378 3206 y Fk(is)43 b(an)g(eigenfunction)e(with)i(eigen-)150 3326 y(value)37 b Fi(r)456 3290 y Fe(\000)p Fl(2)563 3326 y Fk(.)48 b(A)m(l)5 b(l)38 b(other)e(eigenvalues)g(have)g(norm)h(strictly)g(less)f(than)h Fi(r)2886 3290 y Fe(\000)p Fl(2)2992 3326 y Fk(.)49 b(A)m(ny)37 b(c)-5 b(ontinuous)150 3445 y(function)30 b Fi(f)42 b Fk(c)-5 b(an)30 b(b)-5 b(e)31 b(written)g(as)f(a)h(\(p)-5 b(ossibly)30 b(in\014nite\))g(sum)i(of)e(eigenfunctions)f(of)i Fm(\(\010)3441 3409 y Fe(\003)3486 3445 y Fm(\))3525 3409 y Fe(j)p Fh(q)s Fe(\000)p Fh(p)p Fe(j)3719 3445 y Fk(,)150 3565 y(such)36 b(that)f(a)h(subse)-5 b(quenc)g(e)35 b(of)g(p)-5 b(artial)35 b(sums)h(c)-5 b(onver)g(ges)35 b(uniformly)h(to)g Fi(f)11 b Fk(.)150 3714 y Fn(Remark:)53 b Fm(If)38 b Fi(p)d(<)g(q)t Fm(,)j(it)f(turns)h(out)f(that)h(there)g (are)f(a)h(n)m(um)m(b)s(er)g(of)f(functions)i Fi(\020)44 b Fm(for)38 b(whic)m(h)150 3834 y(\(\010)261 3798 y Fe(\003)306 3834 y Fm(\))345 3798 y Fe(j)p Fh(q)s Fe(\000)p Fh(p)p Fe(j)540 3834 y Fi(\020)i Fm(=)33 b(0,)j(but)h(\010)1106 3798 y Fe(\003)1152 3834 y Fi(\020)j Fj(6)p Fm(=)33 b(0.)53 b(Th)m(us,)38 b(to)e(ac)m(hiev)m(e)h(a)f(basis)g(for)h(the)g(space)g (of)g(con)m(tin)m(uous)150 3953 y(functions,)i(w)m(e)f(m)m(ust)f(use)i (eigenfunctions)g(of)e(\(\010)2074 3917 y Fe(\003)2120 3953 y Fm(\))2159 3917 y Fe(j)p Fh(q)s Fe(\000)p Fh(p)p Fe(j)2391 3953 y Fm(rather)g(than)h(just)g(eigenfunctions)150 4073 y(of)33 b(\010)335 4037 y Fe(\003)381 4073 y Fm(.)349 4222 y(Giv)m(en)h(the)f(lemma,)e(w)m(e)j(write)1683 4496 y Fi(f)k Fm(=)1908 4371 y Fe(1)1876 4401 y Fc(X)1883 4613 y Fh(i)p Fl(=0)2036 4496 y Fi(c)2079 4511 y Fh(i)2112 4496 y Fi(\020)2156 4511 y Fh(i)2189 4496 y Fi(:)1278 b Fm(\(4)p Fi(:)p Fm(14\))150 4782 y(Since)44 b(a)f(subsequence)j(of)d (the)g(partial)g(sums)g(con)m(v)m(erges)h(uniformly)-8 b(,)44 b(and)f(since)h Fi(\020)3425 4797 y Fl(0)3513 4782 y Fm(has)f(a)150 4901 y(p)s(ositiv)m(e)33 b(minim)m(um,)d(for)j (eac)m(h)h Fi(\017)29 b(>)f Fm(0)33 b(w)m(e)g(can)h(\014nd)g(an)g(in)m (teger)f Fi(N)44 b Fm(suc)m(h)35 b(that)777 5192 y Fi(f)837 5151 y Fe(\000)931 5192 y Fj(\021)1037 5022 y Fc( )1152 5068 y Fh(N)1115 5098 y Fc(X)1123 5310 y Fh(i)p Fl(=0)1276 5192 y Fi(c)1319 5207 y Fh(k)1368 5192 y Fi(\020)1412 5207 y Fh(i)1445 5022 y Fc(!)1546 5192 y Fj(\000)22 b Fi(\017\020)1729 5207 y Fl(0)1802 5192 y Fi(<)28 b(f)38 b(<)2099 5022 y Fc( )2215 5068 y Fh(N)2178 5098 y Fc(X)2186 5310 y Fh(i)p Fl(=0)2339 5192 y Fi(c)2382 5207 y Fh(k)2431 5192 y Fi(\020)2475 5207 y Fh(i)2508 5022 y Fc(!)2609 5192 y Fm(+)22 b Fi(\017\020)2792 5207 y Fl(0)2865 5192 y Fj(\021)28 b Fi(f)3030 5151 y Fl(+)3095 5192 y Fi(;)372 b Fm(\(4)p Fi(:)p Fm(15\))1900 5519 y(18)p eop %%Page: 19 19 19 18 bop 150 100 a Fm(where)28 b(eac)m(h)g Fi(\020)694 115 y Fh(i)754 100 y Fm(is)f(an)h(eigenfunction)g(with)f(eigen)m(v)-6 b(alue)28 b Fi(\025)2335 115 y Fh(i)2368 100 y Fm(.)42 b(\(Strictly)26 b(sp)s(eaking,)i Fi(\020)3305 115 y Fh(i)3365 100 y Fm(is)f(merely)150 219 y(an)37 b(eigenfunction)h(of)f(\(\010)1135 183 y Fe(\003)1181 219 y Fm(\))1220 183 y Fe(j)p Fh(q)s Fe(\000)p Fh(p)p Fe(j)1414 219 y Fm(,)h(not)e(necessarily)i(of)f(\010) 2350 183 y Fe(\003)2395 219 y Fm(,)h(but)f(this)g(distinction)g(mak)m (es)f(no)150 339 y(di\013erence\).)45 b(Since)34 b Fj(j)p Fi(r)1009 303 y Fe(\000)p Fl(2)1116 339 y Fi(\025)1174 354 y Fh(i)1207 339 y Fj(j)27 b Fi(<)i Fm(1)k(for)g(all)f Fi(i)c(>)h Fm(0,)j(for)i Fi(n)27 b(>)h Fj(j)p Fi(p)22 b Fj(\000)h Fi(q)t Fj(j)33 b Fm(w)m(e)h(ha)m(v)m(e)371 745 y(\()p Fi(c)453 760 y Fl(0)520 745 y Fj(\000)23 b Fi(\017)p Fm(\))p Fi(\020)743 760 y Fl(0)809 745 y Fm(+)946 620 y Fh(N)909 650 y Fc(X)916 863 y Fh(i)p Fl(=1)1069 745 y Fi(c)1112 760 y Fh(k)1162 745 y Fm(\()p Fi(r)1249 704 y Fl(2)1293 745 y Fi(\025)1351 760 y Fh(i)1384 745 y Fm(\))1423 704 y Fh(n)1477 745 y Fi(\020)1521 760 y Fh(i)1581 745 y Fj(\024)29 b Fi(r)1735 704 y Fl(2)p Fh(n)1828 745 y Fi(f)1877 760 y Fh(n)1959 745 y Fj(\024)f Fm(\()p Fi(c)2146 760 y Fl(0)2213 745 y Fm(+)22 b Fi(\017)p Fm(\))p Fi(\020)2435 760 y Fl(0)2502 745 y Fm(+)2638 620 y Fh(N)2602 650 y Fc(X)2609 863 y Fh(i)p Fl(=1)2762 745 y Fi(c)2805 760 y Fh(k)2854 745 y Fm(\()p Fi(r)2941 704 y Fl(2)2985 745 y Fi(\025)3043 760 y Fh(i)3077 745 y Fm(\))3116 704 y Fh(n)3170 745 y Fi(\020)3214 760 y Fh(i)3246 745 y Fi(:)221 b Fm(\(4)p Fi(:)p Fm(16\))150 1146 y(As)27 b Fi(n)g Fj(!)h(1)p Fm(,)f(the)g(left)g(hand)h(side)g(con)m(v)m(erges)g (uniformly)e(to)g(\()p Fi(c)2516 1161 y Fl(0)2570 1146 y Fj(\000)10 b Fi(\017)p Fm(\))p Fi(\020)2780 1161 y Fl(0)2825 1146 y Fm(,)27 b(while)g(the)g(righ)m(t)g(hand)150 1266 y(side)42 b(con)m(v)m(erges)g(uniformly)e(to)g(\()p Fi(c)1485 1281 y Fl(0)1557 1266 y Fm(+)28 b Fi(\017)p Fm(\))p Fi(\020)1785 1281 y Fl(0)1830 1266 y Fm(.)67 b(Since)42 b Fi(\017)f Fm(is)g(arbitrary)-8 b(,)42 b Fi(r)2884 1230 y Fl(2)p Fh(n)2977 1266 y Fi(f)3026 1281 y Fh(n)3121 1266 y Fm(m)m(ust)f(con)m(v)m(erge)150 1385 y(uniformly)32 b(to)h Fi(c)769 1400 y Fl(0)814 1385 y Fi(\020)858 1400 y Fl(0)902 1385 y Fm(.)349 1535 y(All)39 b(that)h(remains)f(is)h(to)g(compute)g Fi(c)1810 1550 y Fl(0)1894 1535 y Fm(in)g(terms)g(of)g Fi(f)11 b Fm(.)64 b(Since)41 b Fi(\020)2885 1550 y Fl(0)2969 1535 y Fm(is)f(in)m(v)-6 b(arian)m(t)40 b(under)150 1654 y(rotation)25 b(and)h(re\015ection,)i Fi(c)1208 1669 y Fl(0)1279 1654 y Fm(m)m(ust)d(b)s(e)i(of)f(the)g(form) 2145 1580 y Fc(P)2250 1684 y Fh(k)2316 1654 y Fi(d)2368 1669 y Fh(k)2433 1574 y Fc(R)2516 1654 y Fi(f)11 b Fm(\()p Fi(k)s(;)17 b Fm(\012\))p Fi(d)p Fm(\012)24 b(for)i(some)g(univ)m (ersal)150 1774 y(constan)m(ts)38 b Fi(d)647 1789 y Fh(k)696 1774 y Fm(.)54 b(By)36 b(comparing)g(c)m(haracteristic)h(functions)h (of)f(di\013eren)m(t)h(sizes,)g(w)m(e)g(see)f(that)150 1894 y(the)45 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b(daugh)m(ter)i(of)g(eac)m(h)g(sub)s(divided)g(tile,)e(the)h(im)m (balance)g(b)s(et)m(w)m(een)h(p)s(ositiv)m(e)f(and)g(negativ)m(e)150 2069 y(orien)m(tation)i(in)h(the)h(larger)e(sizes)i(neither)g(gro)m(ws) f(nor)g(shrinks,)g(but)h(oscillates)e(with)h(p)s(erio)s(d)150 2188 y(2)p Fi(p)p Fm(.)44 b(These)34 b(corresp)s(ond)h(to)e(the)g(ro)s (ots)g(of)g Fi(\025)1835 2152 y Fh(p)1904 2188 y Fm(+)22 b(1.)349 2338 y(If)37 b Fi(p)c(>)h(q)t Fm(,)j(then)g Fi(p)p Fm(\()p Fi(\025)p Fm(\))g(factorizes)g(as)f Fi(p)p Fm(\()p Fi(\025)p Fm(\))d(=)h(\()p Fi(\025)2208 2302 y Fh(p)2278 2338 y Fj(\000)25 b Fm(1)f Fj(\000)h Fm(4)p Fi(\025)2664 2302 y Fh(p)p Fe(\000)p Fh(q)2811 2338 y Fm(\)\()p Fi(\025)2947 2302 y Fh(p)3017 2338 y Fm(+)g(1\).)53 b(Again,)37 b(the)150 2457 y(\014rst)e(factor)h(is)f(the)g(c)m (haracteristic)g(p)s(olynomial)e(of)i Fi(M)11 b Fm(,)35 b(and)g(go)m(v)m(erns)h(the)f(total)f(n)m(um)m(b)s(er)i(of)150 2621 y(tiles)f(of)g(eac)m(h)i(size,)f(with)f(all)f(eigen)m(v)m(ectors)i (ha)m(ving)f Fi(v)g Fm(=)2367 2481 y Fc(\022)2456 2562 y Fm(1)2456 2681 y(1)2523 2481 y Fc(\023)2596 2621 y Fm(.)50 b(The)36 b(largest)f(eigen)m(v)-6 b(alue)36 b(is)150 2797 y Fi(r)198 2761 y Fe(\000)p Fl(2)305 2797 y Fm(,)31 b(with)g(eigen)m(v)m(ector)h Fi(\020)1149 2812 y Fl(0)1193 2797 y Fm(.)43 b(The)32 b(ro)s(ots)f(of)g(\()p Fi(\025)1926 2761 y Fh(p)1990 2797 y Fm(+)19 b(1\))31 b(describ)s(e)h(oscillations)e (in)i(the)f(n)m(um)m(b)s(ers)150 2966 y(of)i(p)s(ositiv)m(ely)f(vs.)44 b(negativ)m(ely)32 b(orien)m(ted)i(tiles)f(and)h(ha)m(v)m(e)g Fi(v)d Fm(=)2553 2826 y Fc(\022)2682 2907 y Fm(1)2643 3026 y Fj(\000)p Fm(1)2787 2826 y Fc(\023)2861 2966 y Fm(.)349 3172 y(Next)38 b(w)m(e)h(consider)g Fi(n)f Fm(o)s(dd.)60 b(Then)39 b(exp\()p Fi(in\031)t Fm(\))c(=)i Fj(\000)p Fm(1,)j(and)f Fi(p)p Fm(\()p Fi(\025)p Fm(\))f(simpli\014es)g(to)f Fi(\025)3531 3136 y Fl(2)p Fh(m)3673 3172 y Fj(\000)150 3292 y Fm(4)p Fi(\025)258 3255 y Fl(2)p Fh(m)p Fe(\000)p Fl(2)p Fh(q)539 3292 y Fj(\000)24 b Fi(\025)698 3255 y Fl(2)p Fh(m)p Fe(\000)p Fl(2)p Fh(p)957 3292 y Fm(.)54 b(This)36 b(is)h(just)g(the)g(c)m(haracteristic)g(p)s(olynomial)d(of)j Fi(M)47 b Fm(applied)36 b(to)h Fi(\025)3678 3255 y Fl(2)3722 3292 y Fm(.)150 3411 y(By)32 b(Theorem)h(6,)g(the)h(largest)e(ro)s(ots) h(ha)m(v)m(e)h Fi(\025)1872 3375 y Fl(2)1944 3411 y Fm(=)28 b Fi(r)2097 3375 y Fe(\000)p Fl(2)2204 3411 y Fm(,)33 b(or)g Fi(\025)28 b Fm(=)g Fj(\006)p Fi(r)2703 3375 y Fe(\000)p Fl(1)2810 3411 y Fm(.)349 3561 y(Finally)c(w)m(e)i(consider)g Fi(n)e Fm(ev)m(en)i(and)g(nonzero.)42 b(Then)26 b Fi(p)p Fm(\()p Fi(\025)p Fm(\))h(=)h Fi(\025)2723 3524 y Fl(2)p Fh(m)2845 3561 y Fj(\000)6 b Fi(\025)2986 3524 y Fl(2)p Fh(m)p Fe(\000)p Fh(q)3203 3561 y Fm(\(4)17 b(cos)o(\()p Fi(n\022)s Fm(\)\))6 b Fj(\000)150 3680 y Fi(\025)208 3644 y Fl(2)p Fh(m)p Fe(\000)p Fl(2)p Fh(p)492 3680 y Fj(\000)27 b Fi(\025)654 3644 y Fl(2)p Fh(m)p Fe(\000)p Fh(p)p Fe(\000)p Fh(q)974 3680 y Fm(\(4)17 b(cos\()p Fi(n\031)t(=)p Fm(2\)\).)59 b(W)-8 b(e)39 b(sho)m(w)g(that)g(all)f(ro)s (ots)g(are)h(smaller)f(than)h Fi(r)3499 3644 y Fe(\000)p Fl(2)3645 3680 y Fm(b)m(y)150 3800 y(the)31 b(argumen)m(t)e(principle.) 44 b(Note)30 b(that)f Fi(\022)k Fm(is)d(an)h(irrational)d(m)m(ultiple)h (of)i Fi(\031)t Fm(,)f(so)g(cos\()p Fi(n\022)s Fm(\))d Fj(6)p Fm(=)h Fj(\006)p Fm(1.)150 3919 y(On)34 b(the)g(circle)f Fj(j)p Fi(\025)p Fj(j)27 b Fm(=)h Fi(r)1042 3883 y Fe(\000)p Fl(2)1182 3919 y Fm(w)m(e)34 b(ha)m(v)m(e)405 4142 y Fj(j)p Fi(\025)491 4101 y Fl(2)p Fh(m)p Fe(\000)p Fh(q)708 4142 y Fm(\(4)17 b(cos)o(\()p Fi(n\022)s Fm(\)\))p Fj(j)k Fm(+)h Fj(j)p Fi(\025)1407 4101 y Fl(2)p Fh(m)p Fe(\000)p Fl(2)p Fh(p)1666 4142 y Fj(j)p Fm(+)p Fj(j)p Fi(\025)1857 4101 y Fl(2)p Fh(m)p Fe(\000)p Fh(p)p Fe(\000)p Fh(q)2177 4142 y Fm(\(4)17 b(cos)o(\()p Fi(n\031)t(=)p Fm(2\)\))p Fj(j)1693 4356 y Fm(=)q Fj(j)p Fi(\025)1857 4314 y Fl(2)p Fh(m)1972 4356 y Fj(j)2017 4215 y Fc(\022)2102 4288 y Fi(b)2145 4252 y Fl(2)2189 4288 y Fj(j)g Fm(cos\()p Fi(n\022)s Fm(\))p Fj(j)p 2102 4333 480 4 v 2297 4424 a Fi(c)2340 4395 y Fl(2)2615 4356 y Fm(+)2727 4288 y Fi(a)2780 4252 y Fl(4)p 2727 4333 98 4 v 2732 4424 a Fi(c)2775 4395 y Fl(4)2858 4356 y Fm(+)2970 4288 y Fi(a)3023 4252 y Fl(2)3067 4288 y Fi(b)3110 4252 y Fl(2)p 2970 4333 185 4 v 3018 4424 a Fi(c)3061 4395 y Fl(4)3167 4215 y Fc(\023)1693 4629 y Fm(=)q Fj(j)p Fi(\025)1857 4587 y Fl(2)p Fh(m)1972 4629 y Fj(j)2012 4561 y Fi(a)2065 4525 y Fl(2)2131 4561 y Fm(+)23 b Fi(b)2274 4525 y Fl(2)2318 4561 y Fj(j)17 b Fm(cos\()p Fi(n\022)s Fm(\))p Fj(j)p 2012 4606 699 4 v 2317 4697 a Fi(c)2360 4668 y Fl(2)1693 4813 y Fi(<)q Fj(j)p Fi(\025)1857 4772 y Fl(2)p Fh(m)1972 4813 y Fj(j)p Fi(:)3495 4477 y Fm(\(4)p Fi(:)p Fm(22\))150 5041 y(Th)m(us)32 b(the)g Fi(\025)628 5005 y Fl(2)p Fh(m)774 5041 y Fm(term)e(of)i Fi(p)p Fm(\()p Fi(\025)p Fm(\))e(dominates)h(on)g(the)g(circle)g Fj(j)p Fi(\025)p Fj(j)d Fm(=)g Fi(r)2671 5005 y Fe(\000)p Fl(2)2777 5041 y Fm(,)k(so)f(the)g(winding)g(of)h(the)150 5160 y(argumen)m(t)i(of)h Fi(p)p Fm(\()p Fi(r)849 5124 y Fe(\000)p Fl(2)972 5160 y Fm(exp\()p Fi(i\013)p Fm(\)\),)f(as)h Fi(\013)f Fm(go)s(es)h(from)f(0)g(to)g(2)p Fi(\031)t Fm(,)f(is)i(2)p Fi(m)p Fm(,)g(and)g(all)e(2)p Fi(m)i Fm(eigen)m(v)-6 b(alues)150 5280 y(of)33 b Fi(E)39 b Fm(lie)33 b(inside)h(the)f(circle)h(of)f(radius)h Fi(r)1685 5244 y Fe(\000)p Fl(2)1791 5280 y Fm(.)1900 5519 y(21)p eop %%Page: 22 22 22 21 bop 349 100 a Fm(This)31 b(completes)f(the)g(analysis)g(of)h(the) f(sp)s(ectrum)h(of)f(\010)2465 63 y Fe(\003)2511 100 y Fm(.)43 b(No)m(w)30 b(w)m(e)h(need)g(only)f(consider)150 219 y(the)44 b(decomp)s(osition)f(in)m(to)g(eigen)m(v)m(ectors.)77 b(Since)44 b Fi(f)54 b Fm(is)44 b(con)m(tin)m(uous,)j(it)c(has)i(an)e (absolutely)150 343 y(summable)d(F)-8 b(ourier)41 b(series.)68 b(So)42 b(w)m(e)f(ma)m(y)f(write)h Fi(f)11 b Fm(\()p Fi(k)s(;)17 b Fm(\012\))39 b(=)2566 268 y Fc(P)2671 372 y Fh(n)2742 343 y Fi(c)2785 358 y Fh(n)2840 343 y Fi( )2909 291 y Fl(\()p Fh(n)p Fl(\))2905 373 y Fh(k)3041 343 y Fm(exp\()p Fi(in\036)p Fm(\),)k(where)150 477 y(the)37 b Fi(c)368 492 y Fh(n)422 477 y Fm('s)g(are)g(absolutely)f(summable)f (and)i Fi( )1913 425 y Fl(\()p Fh(n)p Fl(\))1909 507 y Fh(k)2066 477 y Fm(is)f(a)h(v)m(ector)f(in)h Fb(C)2744 441 y Fl(2)p Fh(m)2902 477 y Fm(with)f(no)h(comp)s(onen)m(t)150 596 y(larger)43 b(than)g(1.)73 b(In)43 b(particular,)i(the)f(partial)e (sums)h(con)m(v)m(erge)h(uniformly)d(to)i Fi(f)11 b Fm(.)73 b(But)42 b(\011)3700 611 y Fh(k)150 716 y Fm(is)f(itself)g(a)h(sum)e (of)i(eigen)m(v)m(ectors)g(of)f Fi(E)1705 680 y Fe(j)p Fh(q)s Fe(\000)p Fh(p)p Fe(j)1899 716 y Fm(,)i(so)f(w)m(e)f(ma)m(y)f (rewrite)h(our)h(sum)f(as)g(a)g(sum)g(of)150 835 y(eigenfunctions)35 b(of)e(\(\010)1025 799 y Fe(\003)1071 835 y Fm(\))1110 799 y Fe(j)p Fh(q)s Fe(\000)p Fh(p)p Fe(j)1304 835 y Fm(.)p 1436 845 60 80 v 150 985 a Fj(x)p Fn(5.)51 b(Irrational)38 b(tilings)349 1134 y Fm(In)45 b(this)f(section)g(w)m(e)h(consider)g (tilings)e Fi(T)14 b(il)r Fm(\()p Fi(z)t Fm(\),)47 b(with)c Fi(z)49 b Fm(irrational.)75 b(The)45 b(analysis)e(is)150 1254 y(formally)21 b(similar)g(to)i(that)f(of)i Fj(x)p Fm(4,)g(except)g(that)e(w)m(e)i(are)f(no)m(w)g(dealing)g(with)g(an)g (in\014nite)g(n)m(um)m(b)s(er)150 1373 y(of)28 b(p)s(ossible)g(sizes.) 42 b(In)28 b(place)g(of)g(the)g(discrete)g(size)g(parameter)f Fi(k)j Fm(w)m(e)e(in)m(tro)s(duce)g(a)g(con)m(tin)m(uous)150 1493 y(size)36 b(parameter)f Fi(s)p Fm(.)51 b(In)35 b(place)h(of)g(the) g(discrete)g(ev)m(olution)f(op)s(erator)h(\010)2898 1457 y Fh(n)2987 1493 y Fm(w)m(e)g(in)m(tro)s(duce)h(a)e(1-)150 1612 y(parameter)26 b(semigroup)g Fi(e)1136 1576 y Fh(tL)1226 1612 y Fm(.)42 b(Although)26 b(the)h(con)m(tin)m(uous)h(case)f(is)g (tec)m(hnically)f(more)g(di\016cult)150 1732 y(than)48 b(the)f(discrete)h(cases,)k(the)c(results)g(are)f(extremely)f(similar.) 84 b(Indeed,)52 b(if)c(one)f(has)h(a)150 1852 y(sequence)h(of)f (rational)f(n)m(um)m(b)s(ers)h Fi(p)1557 1867 y Fh(i)1590 1852 y Fi(=q)1684 1867 y Fh(i)1765 1852 y Fm(con)m(v)m(erging)g(to)f (the)h(irrational)f(n)m(um)m(b)s(er)h Fi(z)t Fm(,)j(then)150 1971 y(the)37 b(statistical)e(prop)s(erties)i(of)g Fi(T)14 b(il)r Fm(\()p Fi(z)t Fm(\))36 b(ma)m(y)g(b)s(e)g(obtained)h(as)g (limits)d(of)j(the)g(corresp)s(onding)150 2091 y(prop)s(erties)g(of)f Fi(T)14 b(il)r Fm(\()p Fi(p)965 2106 y Fh(i)998 2091 y Fi(=q)1092 2106 y Fh(i)1125 2091 y Fm(\).)52 b(\(Note)35 b(that)h(the)g(rev)m(erse)h(do)s(es)g(not)f(hold.)52 b(One)37 b(cannot)f(obtain)150 2210 y(the)e(statistics)e(of)i(a)f (rational)f(tiling)g(b)m(y)h(taking)g(a)g(limit)e(of)i(irrational)f (tilings.\))349 2360 y(As)43 b(alw)m(a)m(ys,)h(w)m(e)f(consider)h(a)e (basic)h(righ)m(t)g(triangle)f Fi(T)2453 2375 y Fl(0)2540 2360 y Fm(with)g(sides)i Fi(a)e Fm(and)h Fi(b)g Fm(and)g(h)m(y-)150 2479 y(p)s(oten)m(use)37 b Fi(c)p Fm(.)50 b(Let)36 b Fi(\013)31 b Fm(=)h(ln\()p Fi(c=a)p Fm(\),)i Fi(\014)j Fm(=)32 b(ln\(2)p Fi(c=b)p Fm(\),)i(and)i(assume)f(that)g Fi(z)h Fm(=)c Fi(\013=\014)40 b Fm(is)c(irrational.)150 2599 y(By)k(a)g(triangle)g(of)g(size)h Fi(s)p Fm(,)h(w)m(e)f(mean)f(a)g (triangle,)h(similar)e(to)h Fi(T)2651 2614 y Fl(0)2696 2599 y Fm(,)i(with)e(h)m(yp)s(oten)m(use)i Fi(ce)3617 2563 y Fe(\000)p Fh(s)3722 2599 y Fm(.)150 2718 y(Note)f(that)g(larger) h(v)-6 b(alues)42 b(of)f Fi(s)h Fm(corresp)s(ond)h(to)e(smaller)f (triangles,)j(just)f(as)g(in)g(the)g(ratio-)150 2838 y(nal)h(case,)j(where)e(the)g(size)f Fi(D)1337 2853 y Fh(k)1430 2838 y Fm(of)g(triangles)g(decreased)i(with)e Fi(k)s Fm(.)74 b(In)43 b(our)h(tiling)e(the)h(size)150 2957 y(parameter)33 b Fi(s)f Fm(will)h(tak)m(e)f(v)-6 b(alues)34 b(in)f([0)p Fi(;)17 b(\026)p Fm(\),)32 b(where)i Fi(\026)28 b Fm(=)g(max)n(\()p Fi(\013;)17 b(\014)5 b Fm(\).)349 3107 y(W)-8 b(e)44 b(no)m(w)h(describ)s(e)g(a)e(semigroup)h (similar)e(to)h(\010)2269 3071 y Fh(n)2324 3107 y Fm(.)75 b(Let)45 b Fi(S)k Fm(b)s(e)44 b(a)g(collection)g(of)g(tiles,)150 3226 y(all)h(with)h(size)h(in)f([0)p Fi(;)17 b(\026)p Fm(\).)82 b(Expand)46 b(this)g(collection)g(b)m(y)h(a)f(linear)g (factor)g Fi(e)3131 3190 y Fh(t)3166 3226 y Fm(,)k(resulting)c(in)150 3346 y(triangles)40 b(with)g(sizes)i(in)e([)p Fj(\000)p Fi(t;)17 b(\026)27 b Fj(\000)g Fi(t)p Fm(\).)65 b(Then)42 b(sub)s(divide)f(the)g(largest)f(triangle,)h(sub)s(divide)150 3466 y(the)35 b(largest)f(remaining)f(triangle,)h(and)h(so)g(on,)f(un)m (til)h(all)e(triangles)i(ha)m(v)m(e)g(non-negativ)m(e)f(size)150 3585 y(parameter.)42 b(By)30 b(Lemmas)f(1)h(and)g(2,)h(this)f(o)s (ccurs)h(in)f(a)g(\014nite)h(n)m(um)m(b)s(er)g(of)f(steps,)i(and)e (results)150 3705 y(in)j(a)g(collection)g(of)g(tiles)g(with)g(sizes)h (in)g([0)p Fi(;)17 b(\026)p Fm(\).)42 b(This)34 b(collection)e(is)i Fi(e)2797 3668 y Fh(tL)2887 3705 y Fi(S)6 b Fm(.)349 3854 y(The)37 b(semigroup)f Fi(e)1083 3818 y Fh(tL)1210 3854 y Fm(naturally)f(acts)i(on)f(the)h(distribution)f(of)h(sizes.)54 b(The)37 b(prop)s(erties)150 3974 y(of)31 b(this)g(action)f(are)g 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b(eigenvalues)e(have)g(r)-5 b(e)g(al)39 b(p)-5 b(art)40 b(gr)-5 b(e)g(ater)39 b(or)g(e)-5 b(qual)40 b(to)g(the)f(r)-5 b(e)g(al)40 b(r)-5 b(o)g(ot)40 b(of)349 5160 y Fi(e)395 5124 y Fh(\014)t(\025)511 5160 y Fm(+)14 b Fi(e)648 5124 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)940 5160 y Fj(\000)g Fm(4)p Fk(,)32 b(while)g(if)f Fi(\013)d(>)g(\014)5 b Fk(,)33 b(al)5 b(l)32 b(eigenvalues)g(have)f(r) -5 b(e)g(al)32 b(p)-5 b(art)32 b(gr)-5 b(e)g(ater)31 b(or)h(e)-5 b(qual)349 5280 y(to)36 b(the)f(r)-5 b(e)g(al)36 b(r)-5 b(o)g(ot)35 b(of)h Fi(e)1190 5244 y Fh(\013\025)1316 5280 y Fm(+)23 b(4)p Fi(e)1512 5244 y Fl(\()p Fh(\013)p Fe(\000)p Fh(\014)t Fl(\))p Fh(\025)1811 5280 y Fj(\000)g Fm(1)p Fk(.)1900 5519 y Fm(22)p eop %%Page: 23 23 23 22 bop 207 100 a Fk(3\))50 b(The)36 b(eigenfunction)e Fi( )t Fm(\()p Fi(s)p Fm(\))p Fk(,)g(for)h(\014xe)-5 b(d)36 b(eigenvalue)f Fi(\025)p Fk(,)g(take)g(the)h(form)981 328 y Fi( )t Fm(\()p Fi(s)p Fm(\))26 b(=)j Fi(e)1353 287 y Fh(\025s)1465 328 y Fj(\000)22 b Fi(e)1610 287 y Fh(\025)p Fl(\()p Fh(s)p Fe(\000)p Fh(\013)p Fl(\))1877 328 y Fi(h)p Fm(\()p Fi(s)g Fj(\000)h Fi(\013)p Fm(\))f Fj(\000)g Fm(4)p Fi(e)2462 287 y Fh(\025)p Fl(\()p Fh(s)p Fe(\000)p Fh(\014)t Fl(\))2726 328 y Fi(h)p Fm(\()p Fi(s)g Fj(\000)g Fi(\014)5 b Fm(\))p Fi(;)427 b Fm(\(5)p Fi(:)p Fm(2\))349 557 y Fk(wher)-5 b(e)36 b Fi(h)p Fm(\()p Fi(x)p Fm(\))f Fk(is)g(the)g(He)-5 b(avyside)35 b(function)1546 810 y Fi(h)p Fm(\()p Fi(x)p Fm(\))27 b(=)1870 699 y Fc(n)1953 760 y Fm(1)102 b Fk(if)35 b Fi(x)27 b Fj(\025)i Fm(0)p Fk(;)1953 879 y Fm(0)102 b Fk(otherwise.)3545 810 y Fm(\(5)p Fi(:)p Fm(3\))207 1093 y Fk(4\))50 b(Given)30 b(an)h(interval)g Fi(I)k Fj(\032)28 b Fm([0)p Fi(;)17 b(\026)p Fm(\))p Fk(,)30 b(the)h(numb)-5 b(er)31 b(of)g(tiles)f(with)h(size)f(in)g Fi(I)39 b Fk(is)30 b(asymptotic)-5 b(al)5 b(ly)349 1213 y(a)36 b(fr)-5 b(action)1286 1300 y Fm(1)p 1242 1345 138 4 v 1242 1436 a(2)p Fi(c)1335 1407 y Fl(2)1408 1232 y Fc(Z)1463 1458 y Fh(I)1526 1368 y Fi(ds)p 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2544 y Fn(Corollary:)56 b Fk(L)-5 b(et)41 b Fi(T)55 b Fk(b)-5 b(e)40 b(a)h(tile)f(of)g(size)g Fi(s)p Fk(.)60 b(L)-5 b(et)42 b Fi(N)2110 2559 y Fh(t)2145 2544 y Fm(\()p Fi(s)p Fm(\))e Fk(b)-5 b(e)40 b(the)h(numb)-5 b(er)41 b(of)f(tiles)g(in)h Fi(e)3480 2508 y Fh(tL)3570 2544 y Fm(\()p Fi(T)14 b Fm(\))p Fk(.)150 2664 y(Then)1353 2812 y Fm(lim)1327 2872 y Fh(t)p Fe(!1)1533 2812 y Fi(e)1579 2771 y Fe(\000)p Fl(2)p Fh(t)1716 2812 y Fi(N)1796 2827 y Fh(t)1832 2812 y Fm(\()p Fi(s)p Fm(\))26 b(=)2152 2745 y(2)p Fi(c)2245 2709 y Fl(2)2290 2745 y Fi(e)2336 2709 y Fe(\000)p Fl(2)p Fh(s)p 2101 2789 V 2101 2880 a Fi(a)2154 2852 y Fl(2)2198 2880 y Fi(\013)c Fm(+)h Fi(b)2427 2852 y Fl(2)2471 2880 y Fi(\014)2545 2812 y(:)972 b Fm(\(5)p Fi(:)p Fm(6\))150 3146 y Fk(Pr)-5 b(o)g(of)38 b(of)h(Cor)-5 b(ol)5 b(lary:)54 b Fm(F)-8 b(rom)35 b(the)j(distribution)f(\(5.4\),)g (w)m(e)h(compute)f(the)h(a)m(v)m(erage)f(area)h(p)s(er)150 3266 y(tile)45 b(to)g(b)s(e)h Fi(ab)p Fm(\()p Fi(a)804 3230 y Fl(2)847 3266 y Fi(\013)31 b Fm(+)f Fi(b)1092 3230 y Fl(2)1137 3266 y Fi(\014)5 b Fm(\))p Fi(=)p Fm(4)p Fi(c)1380 3230 y Fl(2)1425 3266 y Fm(.)80 b(A)45 b(tile)g(of)h(size)g Fi(s)p Fm(,)i(rescaled)e(b)m(y)g(a)f(factor)h Fi(e)3266 3230 y Fh(t)3301 3266 y Fm(,)i(has)e(area)150 3385 y Fi(abe)292 3349 y Fl(2)p Fh(t)p Fe(\000)p Fl(2)p Fh(s)506 3385 y Fi(=)p Fm(2.)e(Dividing)32 b(b)m(y)i(the)f(area)g(p)s(er)h(unit) f(tile)g(w)m(e)g(obtain)g(\(5.6\).)p 2959 3395 60 80 v 150 3535 a Fk(Pr)-5 b(o)g(of)32 b(of)h(The)-5 b(or)g(em)33 b(8:)43 b Fm(F)-8 b(or)30 b(0)e Fi(<)g(t)f(<)i Fm(min)o(\()p Fi(\013;)17 b(\014)5 b(;)17 b Fj(j)p Fi(\013)f Fj(\000)h Fi(\014)5 b Fj(j)p Fm(\),)31 b(the)g(action)g(of)g Fi(e)3026 3499 y Fh(tL)3146 3535 y Fm(on)g(p)s(opulation)150 3654 y(distribution)i(functions)i(is)621 3887 y Fc(\000)667 3967 y Fi(e)713 3926 y Fh(tL)803 3967 y Fi( )872 3887 y Fc(\001)934 3967 y Fm(\()p Fi(s)p Fm(\))27 b(=)1191 3763 y Fc(8)1191 3853 y(<)1191 4032 y(:)1296 3848 y Fi( )t Fm(\()p Fi(s)21 b Fm(+)i Fi(t)p 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y(oscillations)e(at)h Fi(\013)g Fm(and)h Fi(\014)39 b Fm(are)33 b(giv)m(en)g(b)m(y)797 4782 y Fi( )t Fm(\()p Fi(\013)969 4741 y Fe(\000)1036 4782 y Fm(\))22 b Fj(\000)g Fi( )t Fm(\()p Fi(\013)1368 4741 y Fl(+)1434 4782 y Fm(\))27 b(=)h Fi(f)11 b Fm(\(0\);)215 b Fi( )t Fm(\()p Fi(\014)2205 4741 y Fe(\000)2272 4782 y Fm(\))21 b Fj(\000)i Fi( )t Fm(\()p Fi(\014)2601 4741 y Fl(+)2667 4782 y Fm(\))k(=)i(4)p Fi(f)11 b Fm(\(0\))p Fi(:)440 b Fm(\(5)p Fi(:)p Fm(8\))349 5041 y(Setting)30 b Fi(e)735 5005 y Fh(tL)825 5041 y Fi( )h Fm(=)e Fi(e)1073 5005 y Fh(t\025)1155 5041 y Fi( )34 b Fm(w)m(e)c(see)h(that)f(the)g (eigenfunction)i Fi( )t Fm(\()p Fi(s)p Fm(\))c(m)m(ust)i(equal)g Fi(e)3317 5005 y Fh(\025s)3437 5041 y Fm(times)f(a)150 5160 y(piecewise)36 b(constan)m(t)f(function)h(with)f(discon)m(tin)m (uities)h(at)e(0)p Fi(;)17 b(\013;)g(\014)5 b Fm(.)48 b(Applying)34 b(the)i(b)s(oundary)150 5280 y(conditions)42 b(\(5.8\),)g(w)m(e)g(obtain)f(the)h(eigenfunction)h(\(5.2\).)68 b(F)-8 b(or)41 b Fi(s)g(>)h(\026)p Fm(,)i Fi( )t Fm(\()p Fi(s)p Fm(\))c(then)i(equals)1900 5519 y(23)p eop %%Page: 24 24 24 23 bop 150 100 a Fi(e)196 63 y Fl(\()p Fh(s)p Fe(\000)p Fh(\026)p Fl(\))p Fh(\025)493 100 y Fm(times)33 b Fi(p)p Fm(\()p Fi(\025)p Fm(\).)46 b(The)35 b(v)-6 b(anishing)34 b(of)g Fi( )t Fm(\()p Fi(s)p Fm(\))f(for)h Fi(s)29 b(>)g(\026)34 b Fm(is)g(equiv)-6 b(alen)m(t)34 b(to)g(the)g(eigen)m(v)-6 b(alue)150 219 y(equation)37 b Fi(p)p Fm(\()p Fi(\025)p Fm(\))d(=)h(0.)57 b(Th)m(us)38 b(eigenfunctions)h(satisfying)e(the)h(b) s(oundary)g(conditions)f(are)h(in)150 339 y(1{1)33 b(corresp)s(ondence) i(with)e(ro)s(ots)g(of)g Fi(p)p Fm(\()p Fi(\025)p Fm(\),)g(with)g(the)h (eigenfunctions)g(giv)m(en)g(b)m(y)f(\(5.2\).)349 488 y(Supp)s(ose)i Fi(\013)28 b(<)g(\014)5 b Fm(,)33 b(in)h(whic)m(h)g (case)g Fi(p)f Fm(tak)m(es)g(the)h(form)1395 723 y Fi(p)p Fm(\()p Fi(\025)p Fm(\))27 b(=)h Fi(e)1759 682 y Fh(\014)t(\025)1883 723 y Fj(\000)22 b Fi(e)2028 682 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)2328 723 y Fj(\000)h Fm(4)p Fi(:)1039 b Fm(\(5)p Fi(:)p Fm(9\))150 964 y Fi(\025)37 b Fm(=)h(2)h(is)g(a)g(ro)s(ot,)g(since)h Fi(p)p Fm(\(2\))d(=)h Fi(e)1518 928 y Fl(2)p Fh(\014)1637 964 y Fj(\000)27 b Fi(e)1787 928 y Fl(2\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))2083 964 y Fj(\000)g Fm(4)37 b(=)2400 925 y Fl(4)p Fh(c)2476 895 y Ff(2)p 2400 941 115 4 v 2421 999 a Fh(b)2456 979 y Ff(2)2553 964 y Fj(\000)2668 925 y Fl(4)p Fh(a)2751 895 y Ff(2)p 2668 941 122 4 v 2692 999 a Fh(b)2727 979 y Ff(2)2828 964 y Fj(\000)26 b Fm(4)37 b(=)h(0.)61 b(This)40 b(is)f(the)150 1084 y(only)c(real)g(ro)s(ot,)g(insofar)h(as)g Fi(p)p Fm(\()p Fi(\025)p Fm(\))f(is)h(an)g(increasing)g(function)h(of)e Fi(\025)h Fm(for)g Fi(\025)c(>)g Fm(0,)k(and)g Fi(p)p Fm(\()p Fi(\025)p Fm(\))f(is)150 1203 y(negativ)m(e)e(for)g Fi(\025)28 b Fj(\024)g Fm(0.)44 b(In)34 b(particular,)e Fi(p)p Fm(\()p Fi(\025)p Fm(\))c Fi(<)g Fm(0)33 b(implies)f(that)h Fi(\025)28 b(<)g Fm(2.)349 1353 y(No)m(w)34 b(consider)i(complex)d(ro)s (ots)h Fi(\025)c Fm(=)g Fi(\025)1865 1368 y Fh(R)1954 1353 y Fm(+)23 b Fi(i\025)2146 1368 y Fh(I)2192 1353 y Fm(.)48 b(If)35 b Fi(\025)2427 1368 y Fh(I)2502 1353 y Fj(6)p Fm(=)c(0,)j Fi(e)2768 1317 y Fh(\014)t(\025)2904 1353 y Fm(and)h Fi(e)3145 1317 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)3457 1353 y Fm(cannot)150 1472 y(b)s(oth)f(b)s(e)h(real,)e(insofar)i(as)f Fi(\014)40 b Fm(is)34 b(not)g(a)g(rational)e(m)m(ultiple)i(of)g Fi(\014)28 b Fj(\000)23 b Fi(\013)p Fm(.)47 b(Th)m(us)35 b Fj(j)p Fi(e)3180 1436 y Fh(\014)t(\025)3281 1472 y Fj(j)p Fm(,)f Fj(j)p Fi(e)3445 1436 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)3722 1472 y Fj(j)150 1592 y Fm(and)g(4)f(satisfy)g(a)g(strict)g(triangle)f(inequalit)m(y)-8 b(.)43 b(In)33 b(particular,)1215 1826 y(0)28 b Fi(>)g Fj(j)p Fi(e)1472 1785 y Fh(\014)t(\025)1573 1826 y Fj(j)22 b(\000)g(j)p Fi(e)1796 1785 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)2074 1826 y Fj(j)g(\000)g Fm(4)28 b(=)g Fi(p)p Fm(\()p Fi(\025)2553 1841 y Fh(R)2618 1826 y Fm(\))p Fi(;)810 b Fm(\(5)p Fi(:)p Fm(10\))150 2061 y(so)33 b Fi(\025)330 2076 y Fh(R)423 2061 y Fi(<)c Fm(2.)43 b(Also,)902 2296 y(0)28 b Fi(<)g Fj(j)p Fi(e)1159 2255 y Fh(\014)t(\025)1260 2296 y Fj(j)22 b Fm(+)g Fj(j)p Fi(e)1483 2255 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)1761 2296 y Fj(j)g(\000)g Fm(4)28 b(=)g Fi(e)2139 2255 y Fh(\014)t(\025)2235 2265 y Fg(R)2319 2296 y Fm(+)22 b Fi(e)2464 2255 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)2736 2265 y Fg(R)2821 2296 y Fj(\000)g Fm(4)p Fi(;)497 b Fm(\(5)p Fi(:)p Fm(11\))150 2530 y(so)33 b Fi(\025)330 2545 y Fh(R)429 2530 y Fm(is)g(greater)g (than)h(the)f(real)g(ro)s(ot)g(of)g Fi(e)1838 2494 y Fh(\014)t(\025)1962 2530 y Fm(+)22 b Fi(e)2107 2494 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)2407 2530 y Fj(\000)h Fm(4.)349 2680 y(No)m(w)k(w)m(e)g(exclude)g(the)g(p)s (ossibilit)m(y)f(of)g(m)m(ultiple)g(ro)s(ots.)41 b(A)27 b(m)m(ultiple)e(ro)s(ot)h(w)m(ould)h(require)150 2799 y Fi(p)p Fm(\()p Fi(\025)p Fm(\))g(=)i Fi(p)519 2763 y Fe(0)547 2799 y Fm(\()p Fi(\025)p Fm(\))e(=)h(0.)41 b(Supp)s(ose)25 b(0)j(=)g Fi(p)1548 2763 y Fe(0)1576 2799 y Fm(\()p Fi(\025)p Fm(\))f(=)i Fi(\014)5 b(e)1952 2763 y Fh(\014)t(\025)2057 2799 y Fm(+)t(\()p Fi(\014)k Fj(\000)t Fi(\013)p Fm(\))p Fi(e)2472 2763 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)2750 2799 y Fm(.)41 b(Then)25 b Fi(e)3116 2763 y Fh(\014)t(\025)3242 2799 y Fm(and)f Fi(e)3472 2763 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)150 2919 y Fm(m)m(ust)i(ha)m(v)m(e)h(the)f (same)g(phase,)i(and)f(their)f(di\013erence)i(m)m(ust)e(also)g(ha)m(v)m (e)g(that)g(phase.)43 b(Ho)m(w)m(ev)m(er,)150 3038 y(if)33 b Fi(\025)g Fm(is)g(not)g(real,)f Fi(e)876 3002 y Fh(\014)t(\025)1011 3038 y Fm(and)h Fi(e)1250 3002 y Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)1561 3038 y Fm(cannot)g(b)s(oth)g(b)s(e)h (real,)e(so)h(their)g(di\013erence)i(is)e(not)g(real,)150 3158 y(so)h(their)f(di\013erence)j(is)d(not)h(4.)45 b(Th)m(us)34 b Fi(p)1661 3122 y Fe(0)1690 3158 y Fm(\()p Fi(\025)p Fm(\))28 b(=)g(0)34 b(implies)e(that)i Fi(p)p Fm(\()p Fi(\025)p Fm(\))28 b Fj(6)p Fm(=)h(0,)k(and)h(there)g(are)g(no)150 3278 y(m)m(ultiple)28 b(ro)s(ots)h(a)m(w)m(a)m(y)f(from)h(the)g(real)g (axis.)42 b(On)30 b(the)f(real)g(axis,)g(the)g(only)g(ro)s(ot)f(is)h Fi(\025)f Fm(=)g(2,)h(and)150 3397 y(w)m(e)34 b(ha)m(v)m(e)f(already)g (seen)i(that)d Fi(p)1360 3361 y Fe(0)1388 3397 y Fm(\(2\))h(is)g(p)s (ositiv)m(e,)f(not)i(zero.)349 3546 y(This)j(establishes)h(statemen)m (ts)e(1{3)g(for)h(the)g(case)h Fi(\013)33 b(<)h(\014)5 b Fm(.)55 b(The)37 b(argumen)m(t)f(for)h Fi(\013)d(>)f(\014)150 3666 y Fm(is)g(almost)f(iden)m(tical,)h(and)g(is)h(not)f(rep)s(eated.) 349 3815 y(No)m(w)39 b(supp)s(ose)h(that)f(w)m(e)g(initially)e(ha)m(v)m (e)j(a)e(p)s(opulation)h(distribution)g(that)g(is)f(a)h(linear)150 3935 y(com)m(bination)e(of)h(the)g(eigenfunctions)h Fi( )1736 3950 y Fh(\025)1788 3935 y Fm(.)57 b(Applying)37 b Fi(e)2355 3899 y Fh(tL)2483 3935 y Fm(to)g(the)h(system,)g(the)g(total)e(area)150 4055 y(gro)m(ws)27 b(as)g Fi(e)586 4018 y Fl(2)p Fh(t)661 4055 y Fm(,)h(so)e(the)h(n)m(um)m(b)s(er)h(of)f(tiles)f(is)h(b)s (ounded,)i(b)s(oth)e(ab)s(o)m(v)m(e)g(and)g(b)s(elo)m(w,)h(b)m(y)e(a)h (m)m(ultiple)150 4174 y(of)36 b Fi(e)312 4138 y Fl(2)p Fh(t)387 4174 y Fm(.)51 b(Applying)35 b Fi(e)946 4138 y Fh(tL)1071 4174 y Fm(and)i(dividing)d(b)m(y)i(the)g(n)m(um)m(b)s(er)g (of)g(tiles)f(damps)g(out)h(all)f(the)g(mo)s(des)150 4294 y(with)40 b(eigen)m(v)-6 b(alue)41 b(less)g(than)g(2,)h(i.e.)65 b(all)39 b(eigen)m(v)m(ectors)j(other)e(than)h Fi( )2893 4309 y Fl(2)2937 4294 y Fm(.)66 b(In)41 b(this)f(case,)j(the)150 4413 y(\014nal)d(distribution)f(of)h(sizes)g(approac)m(hes)h(a)e(m)m (ultiple)f(of)i Fi( )2470 4428 y Fl(2)2515 4413 y Fm(,)g(and)g (statemen)m(ts)f(4)h(and)g(5)f(of)150 4533 y(the)34 b(theorem)e(follo)m (w.)349 4682 y(Unfortunately)-8 b(,)35 b(w)m(e)h(cannot)f(a)g(priori)g (assume)g(that)g(the)h(initial)e(condition)h(is)g(a)g(linear)150 4802 y(com)m(bination)f(of)i(eigenfunctions)h(of)e Fi(L)p Fm(,)g(or)g(that)g(a)g(test)g(function)i(is)e(a)g(linear)g(com)m (bination)150 4921 y(of)49 b(eigenfunctions)h(of)e(the)h(dual)g(op)s (erator)f Fi(L)1982 4885 y Fe(\003)2027 4921 y Fm(.)90 b Fi(e)2191 4885 y Fh(tL)2329 4921 y Fm(and)49 b Fi(e)2584 4885 y Fh(tL)2669 4855 y Fd(\003)2764 4921 y Fm(are)g(neither)g (\014nite-rank)150 5041 y(op)s(erators)29 b(nor)g(self-adjoin)m(t)h(op) s(erators)e(on)h(a)g(Hilb)s(ert)f(space,)i(so)f(standard)h(theorems)e (ab)s(out)150 5160 y(the)39 b(completeness)h(of)f(a)g(basis)g(is)g (eigenfunctions)i(cannot)e(b)s(e)h(applied.)62 b(In)39 b(principle)h(it)e(is)150 5280 y(p)s(ossible)31 b(for)f(a)g(test)g (function)h Fi(f)40 b Fm(to)30 b(ha)m(v)m(e)h(the)f(prop)s(ert)m(y)g (that)g Fi(e)2595 5244 y Fh(t)p Fl(\()p Fh(L)2711 5214 y Fd(\003)2753 5244 y Fe(\000)p Fl(2\))2891 5280 y Fi(f)41 b Fm(do)s(es)30 b(not)g(con)m(v)m(erge)1900 5519 y(24)p eop %%Page: 25 25 25 24 bop 150 100 a Fm(at)33 b(all.)44 b(W)-8 b(e)34 b(m)m(ust)f(sho)m(w)i(that,)e(when)h Fi(f)39 b Fm(=)29 b Fi(\037)1877 115 y Fh(I)1956 100 y Fm(\(the)34 b(c)m(haracteristic)g (function)h(of)e(an)h(in)m(terv)-6 b(al)150 219 y Fi(I)8 b Fm(\),)32 b Fi(e)347 183 y Fh(t)p Fl(\()p Fh(L)463 153 y Fd(\003)505 183 y Fe(\000)p Fl(2\))643 219 y Fi(f)44 b Fk(do)-5 b(es)32 b Fm(con)m(v)m(erge.)150 369 y Fn(Lemma)g(9:)41 b Fk(Given)30 b(an)h(interval)f Fi(I)35 b Fj(2)28 b Fm([0)p Fi(;)17 b(\026)p Fm(\))p Fk(.)43 b(The)30 b(fr)-5 b(action)30 b(of)g(the)g(ar)-5 b(e)g(a)30 b(of)g Fi(e)3137 332 y Fh(tL)3227 369 y Fm(\()p Fi(T)3324 384 y Fl(0)3369 369 y Fm(\))g Fk(c)-5 b(over)g(e)g(d)150 488 y(by)36 b(tiles)f(with)h(size) e(in)h Fi(I)44 b Fk(appr)-5 b(o)g(aches)34 b(a)h(limit)h(as)g Fi(t)27 b Fj(!)g(1)p Fk(.)349 638 y Fm(Giv)m(en)43 b(this)f(lemma,)g (it)f(follo)m(ws)h(that)g(the)g(distribution)g(of)h(area)f(of)g Fi(e)3127 601 y Fh(tL)3217 638 y Fm(\()p Fi(S)6 b Fm(\),)44 b(for)e(an)m(y)150 757 y(collection)33 b(of)g(tiles)g Fi(S)6 b Fm(,)33 b(approac)m(hes)h(a)f(limit,)e(from)h(whic)m(h)i(it)f (follo)m(ws)g(that)f(the)i(distribution)150 877 y(of)44 b(p)s(opulation)f(of)g Fi(e)954 841 y Fh(tL)1044 877 y Fm(\()p Fi(S)6 b Fm(\))43 b(also)g(approac)m(hes)i(a)e(limit.)72 b(Since)45 b(these)f(limits)e(are)h(in)m(v)-6 b(arian)m(t)150 996 y(under)33 b(further)g(ev)m(olution,)e(and)i(since)f(the)h(total)d (area)i(is)g(prop)s(ortional)f(to)g Fi(e)3117 960 y Fl(2)p Fh(t)3192 996 y Fm(,)h(these)h(limits)150 1116 y(m)m(ust)e(corresp)s (ond)i(to)d(the)i Fi(\025)c Fm(=)g(2)j(eigen)m(v)m(ector)h(of)f Fi(L)p Fm(,)h(hence)g(m)m(ust)f(tak)m(e)g(the)h(form)e(\(5.4\))g(and) 150 1235 y(\(5.5\).)p 487 1245 60 80 v 150 1385 a Fk(Pr)-5 b(o)g(of)45 b(of)g(L)-5 b(emma)47 b(9:)66 b Fm(Assume)45 b Fi(\013)h(<)h(\014)5 b Fm(;)50 b(the)45 b(other)f(case)h(is)g (similar.)76 b(Let)44 b Fi(F)3261 1400 y Fh(I)3307 1385 y Fm(\()p Fi(t)p Fm(\))g(b)s(e)h(the)150 1504 y(fraction)36 b(of)h(area)f(of)g Fi(e)1019 1468 y Fh(tL)1109 1504 y Fm(\()p Fi(T)1206 1519 y Fl(0)1251 1504 y Fm(\))g(co)m(v)m(ered)h(b)m (y)g(tiles)f(with)f(size)i(in)f Fi(I)8 b Fm(.)53 b(Giv)m(en)36 b(an)h Fi(\017)c(>)g Fm(0,)j(w)m(e)h(will)150 1624 y(sho)m(w)30 b(ho)m(w)f(to)g(compute)g(a)g(n)m(um)m(b)s(er)h(suc)m(h)h(that,)e(for)h (all)e Fi(t)i Fm(su\016cien)m(tly)f(large,)h Fi(F)3189 1639 y Fh(I)3234 1624 y Fm(\()p Fi(t)p Fm(\))f(is)g(within)150 1743 y Fi(\017)34 b Fm(of)f(this)g(n)m(um)m(b)s(er.)45 b(Since)34 b(this)f(can)h(b)s(e)g(done)g(for)f(an)m(y)g Fi(\017)p Fm(,)h(lim)2524 1758 y Fh(t)p Fe(!1)2734 1743 y Fi(F)2798 1758 y Fh(I)2844 1743 y Fm(\()p Fi(t)p Fm(\))f(m)m(ust)g (exist.)349 1893 y(It)28 b(su\016ces)i(to)f(sho)m(w)g(that)f(the)h(ev)m (en)m(tual)g(fraction)f(in)h(an)f(in)m(terv)-6 b(al)29 b(of)g(size)g(\001,)g(en)m(tirely)f(in)150 2012 y([0)p Fi(;)17 b(\013)p Fm(\))29 b(or)i(in)g([)p Fi(\013;)17 b(\014)5 b Fm(\),)30 b(and)h(with)g(\001)g(su\016cien)m(tly)g(small,)e (can)i(b)s(e)h(estimated)e(to)g(within)g Fi(O)s Fm(\(\001)3639 1976 y Fl(2)3684 2012 y Fm(\).)150 2132 y(An)m(y)f(larger)h(in)m(terv) -6 b(al)29 b(can)h(b)s(e)g(brok)m(en)g(up)g(in)m(to)g(a)f(\014nite)h(n) m(um)m(b)s(er)g(of)g(suc)m(h)h(small)d(pieces,)j(suc)m(h)150 2252 y(that)366 2177 y Fc(P)504 2252 y Fm(errors)j Fi(<)28 b(\017)p Fm(.)44 b(So)34 b(let)f(us)h(\014x)f(an)g(in)m(terv)-6 b(al)33 b Fi(I)8 b Fm(,)33 b(cen)m(tered)i(at)d Fi(s)2732 2267 y Fl(0)2777 2252 y Fm(,)h(with)g(width)g(\001.)349 2401 y(The)45 b(strategy)f(is)g(this:)67 b(W)-8 b(e)44 b(b)s(egin)h(with)f(an)g(exact)h(expression)g(for)f(the)h(fraction)g (of)150 2521 y(area)d(of)g Fi(e)543 2484 y Fh(tL)633 2521 y Fm(\()p Fi(T)730 2536 y Fl(0)775 2521 y Fm(\))f(represen)m(ted)k (b)m(y)d(tiles)f(of)h(size)h Fi(s)p Fm(.)69 b(W)-8 b(e)42 b(sum)g(this)g(o)m(v)m(er)g Fi(s)g Fj(2)g Fi(I)50 b Fm(to)41 b(get)h(an)150 2640 y(exact)37 b(form)m(ula)f(for)i Fi(F)1007 2655 y Fh(I)1052 2640 y Fm(\()p Fi(t)p Fm(\).)55 b(By)37 b(taking)f(certain)h(limits)e(and)j(replacing)f(certain)g(sums)g(with) 150 2760 y(in)m(tegrals,)52 b(w)m(e)d(obtain)g(an)g(expression)g(that)g (is)f(indep)s(enden)m(t)j(of)e Fi(t)p Fm(.)91 b(In)49 b(the)g(pro)s(cess)h(w)m(e)150 2879 y(in)m(tro)s(duce)33 b(t)m(w)m(o)e(t)m(yp)s(es)h(of)g(errors.)44 b(One)33 b(t)m(yp)s(e)f(can)g(b)s(e)g(made)f(arbitrarily)f(small)g(b)m(y)i (requiring)150 2999 y Fi(t)h Fm(to)g(b)s(e)h(su\016cien)m(tly)f(large.) 44 b(The)34 b(other)f(t)m(yp)s(e)g(is)g Fi(O)s Fm(\(\001)2248 2963 y Fl(2)2293 2999 y Fm(\).)349 3148 y(Ho)m(w)27 b(man)m(y)f (triangles)h(of)h(size)g Fi(s)f Fm(app)s(ear)g(in)h Fi(e)2079 3112 y Fh(tL)2169 3148 y Fm(\()p Fi(T)2266 3163 y Fl(0)2310 3148 y Fm(\)?)42 b(That)27 b(dep)s(ends)j(on)d(whether)h Fi(s)10 b Fm(+)g Fi(t)150 3268 y Fm(can)30 b(b)s(e)h(written)f(as)g Fi(n)987 3283 y Fl(1)1031 3268 y Fi(\013)16 b Fm(+)g Fi(n)1264 3283 y Fl(2)1309 3268 y Fi(\014)35 b Fm(for)c(non-negativ)m (e)f Fi(n)2193 3283 y Fl(1)2268 3268 y Fm(and)g Fi(n)2518 3283 y Fl(2)2563 3268 y Fm(.)42 b(If)31 b Fi(s)16 b Fm(+)g Fi(t)27 b Fm(=)h Fi(n)3114 3283 y Fl(1)3159 3268 y Fi(\013)16 b Fm(+)g Fi(n)3392 3283 y Fl(2)3436 3268 y Fi(\014)5 b Fm(,)31 b(then)150 3387 y(a)42 b(triangle)f(of)h(size)h Fi(s)e Fm(ma)m(y)g(b)s(e)h(obtained)h(b)m(y)f(taking)f(a)g(triangle)h (of)g(size)g Fj(\000)p Fi(t)p Fm(,)j(sub)s(dividing)150 3507 y(it,)e(pic)m(king)e(a)g(daugh)m(ter,)j(sub)s(dividing)e(it,)h (pic)m(king)e(a)g(daugh)m(ter,)j(and)e(so)g(on)g(for)f Fi(n)3468 3522 y Fl(1)3540 3507 y Fm(+)29 b Fi(n)3706 3522 y Fl(2)150 3626 y Fm(sub)s(divisions,)g(with)e(the)g(descen)m(t)j (in)m(v)m(olving)c Fi(n)1947 3641 y Fl(1)2019 3626 y Fm(daugh)m(ters)i(of)g(t)m(yp)s(e)f Fi(A)g Fm(and)h Fi(n)3146 3641 y Fl(2)3218 3626 y Fm(daugh)m(ters)h(of)150 3746 y(t)m(yp)s(e)g Fi(B)5 b Fm(.)42 b(If)29 b Fi(s)f Fj(\025)g Fi(\013)p Fm(,)i(then)f(the)h(last)e(daugh)m(ter)i(m)m(ust)f(b)s(e)g (of)h(t)m(yp)s(e)f Fi(B)5 b Fm(,)29 b(or)f(else)i(after)f Fi(n)3341 3761 y Fl(1)3400 3746 y Fm(+)14 b Fi(n)3551 3761 y Fl(2)3609 3746 y Fj(\000)g Fm(1)150 3866 y(steps)41 b(w)m(e)g(w)m(ould)g(ha)m(v)m(e)f(already)g(obtained)h(a)f(tile)f(of)i (size)g(in)f([0)p Fi(;)17 b(\014)5 b Fm(\),)41 b(and)g(w)m(ould)f(not)h (ha)m(v)m(e)150 3985 y(made)33 b(the)g(\014nal)h(sub)s(division.)44 b(If)33 b Fi(s)28 b(<)g(\013)33 b Fm(there)h(is)f(no)h(suc)m(h)g (constrain)m(t.)45 b(W)-8 b(e)33 b(th)m(us)h(ha)m(v)m(e)170 4224 y(Num)m(b)s(er)f(of)h(tiles)f(of)g(size)h Fi(s)27 b Fm(=)1361 4019 y Fc(8)1361 4109 y(<)1361 4288 y(:)1467 4030 y(\000)1512 4067 y Fh(n)1561 4077 y Ff(1)1600 4067 y Fl(+)p Fh(n)1710 4077 y Ff(2)1587 4145 y Fh(n)1636 4155 y Ff(1)1750 4030 y Fc(\001)1795 4111 y Fm(4)1845 4075 y Fh(n)1894 4085 y Ff(2)2140 4111 y Fm(if)33 b Fi(s)22 b Fm(+)h Fi(t)k Fm(=)h Fi(n)2628 4126 y Fl(1)2673 4111 y Fi(\013)22 b Fm(+)h Fi(n)2919 4126 y Fl(2)2963 4111 y Fi(\014)39 b Fm(and)34 b Fi(s)27 b Fj(2)h Fm([0)p Fi(;)17 b(\013)p Fm(\);)1467 4174 y Fc(\000)1512 4210 y Fh(n)1561 4220 y Ff(1)1600 4210 y Fl(+)p Fh(n)1710 4220 y Ff(2)1750 4210 y Fe(\000)p Fl(1)1638 4289 y Fh(n)1687 4299 y Ff(1)1852 4174 y Fc(\001)1897 4254 y Fm(4)1947 4218 y Fh(n)1996 4228 y Ff(2)2140 4254 y Fm(if)33 b Fi(s)22 b Fm(+)h Fi(t)k Fm(=)h Fi(n)2628 4269 y Fl(1)2673 4254 y Fi(\013)22 b Fm(+)h Fi(n)2919 4269 y Fl(2)2963 4254 y Fi(\014)39 b Fm(and)34 b Fi(s)27 b Fj(2)h Fm([)p Fi(\013;)17 b(\014)5 b Fm(\);)1467 4374 y(0)623 b(otherwise.)3495 4453 y(\(5)p Fi(:)p Fm(12\))150 4572 y(Since)36 b(at)f(eac)m(h)h(division)f(a)g (fraction)g Fi(a)1641 4536 y Fl(2)1686 4572 y Fi(=c)1779 4536 y Fl(2)1859 4572 y Fm(of)g(the)h(area)f(go)s(es)g(in)m(to)g(the)h Fi(A)f Fm(daugh)m(ter,)h(while)150 4692 y(a)49 b(fraction)g Fi(b)676 4656 y Fl(2)721 4692 y Fi(=c)814 4656 y Fl(2)907 4692 y Fm(go)s(es)g(in)m(to)g(the)h Fi(B)j Fm(daugh)m(ters,)h(then)c (the)f(fraction)g(of)h(the)f(total)f(area)150 4811 y(represen)m(ted)36 b(b)m(y)d(tiles)g(of)g(size)h Fi(s)f Fm(is)203 5115 y Fi(F)267 5130 y Fh(s)309 5115 y Fm(\()p Fi(t)p Fm(\))27 b(=)556 4881 y Fc(8)556 4971 y(>)556 5001 y(<)556 5180 y(>)556 5210 y(:)661 4901 y(\000)706 4937 y Fh(n)755 4947 y Ff(1)795 4937 y Fl(+)p Fh(n)905 4947 y Ff(2)781 5016 y Fh(n)830 5026 y Ff(1)944 4901 y Fc(\001)1006 4871 y(\020)1078 4942 y Fh(a)1121 4912 y Ff(2)p 1078 4958 83 4 v 1082 5016 a Fh(c)1118 4996 y Ff(2)1172 4871 y Fc(\021)1231 4891 y Fh(n)1280 4901 y Ff(1)1341 4871 y Fc(\020)1413 4942 y Fh(b)1448 4912 y Ff(2)p 1412 4958 75 4 v 1412 5016 a Fh(c)1448 4996 y Ff(2)1499 4871 y Fc(\021)1558 4891 y Fh(n)1607 4901 y Ff(2)1853 4981 y Fm(if)33 b Fi(s)22 b Fm(+)h Fi(t)k Fm(=)i Fi(n)2342 4996 y Fl(1)2386 4981 y Fi(\013)22 b Fm(+)h Fi(n)2632 4996 y Fl(2)2676 4981 y Fi(\014)39 b Fm(and)34 b Fi(s)27 b Fj(2)h Fm([0)p Fi(;)17 b(\013)p Fm(\);)661 5095 y Fc(\000)706 5132 y Fh(n)755 5142 y Ff(1)795 5132 y Fl(+)p Fh(n)905 5142 y Ff(2)944 5132 y Fe(\000)p Fl(1)832 5210 y Fh(n)881 5220 y Ff(1)1046 5095 y Fc(\001)1108 5065 y(\020)1180 5137 y Fh(a)1223 5107 y Ff(2)p 1180 5153 83 4 v 1184 5210 a Fh(c)1220 5190 y Ff(2)1274 5065 y Fc(\021)1333 5086 y Fh(n)1382 5096 y Ff(1)1443 5065 y Fc(\020)1515 5137 y Fh(b)1550 5107 y Ff(2)p 1514 5153 75 4 v 1514 5210 a Fh(c)1550 5190 y Ff(2)1601 5065 y Fc(\021)1660 5086 y Fh(n)1709 5096 y Ff(2)1853 5176 y Fm(if)33 b Fi(s)22 b Fm(+)h Fi(t)k Fm(=)i Fi(n)2342 5191 y Fl(1)2386 5176 y Fi(\013)22 b Fm(+)h Fi(n)2632 5191 y Fl(2)2676 5176 y Fi(\014)39 b Fm(and)34 b Fi(s)27 b Fj(2)h Fm([)p Fi(\013;)17 b(\014)5 b Fm(\);)661 5320 y(0)1142 b(otherwise.)3495 5115 y(\(5)p Fi(:)p Fm(13\))1900 5519 y(25)p eop %%Page: 26 26 26 25 bop 150 100 a Fm(Note)33 b(that)g Fi(n)667 115 y Fl(2)739 100 y Fm(=)28 b(\()p Fi(s)22 b Fm(+)g Fi(t)g Fj(\000)h Fi(\013n)1333 115 y Fl(1)1377 100 y Fm(\))p Fi(=\014)5 b Fm(.)45 b(No)m(w)33 b(let)293 466 y Fi(f)11 b Fm(\()p Fi(s;)17 b(t;)g(n)p Fm(\))24 b(=)793 232 y Fc(8)793 322 y(>)793 351 y(<)793 531 y(>)793 561 y(:)899 291 y(\000)944 327 y Fl(\()p Fh(s)p Fl(+)p Fh(t)p Fl(+)p Fh(n)p Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\)\))p Fh(=\014)1228 406 y(n)1560 291 y Fc(\001)1623 261 y(\020)1694 332 y Fh(a)1737 302 y Ff(2)p 1694 348 83 4 v 1698 406 a Fh(c)1734 386 y Ff(2)1788 261 y Fc(\021)1848 281 y Fh(n)1919 261 y Fc(\020)1990 332 y Fh(b)2025 302 y Ff(2)p 1990 348 75 4 v 1990 406 a Fh(c)2026 386 y Ff(2)2076 261 y Fc(\021)2136 281 y Fl(\()p Fh(s)p Fl(+)p Fh(t)p Fe(\000)p Fh(\013n)p Fl(\))p Fh(=\014)2796 371 y Fm(if)33 b Fi(s)27 b Fj(2)h Fm([0)p Fi(;)17 b(\013)p Fm(\);)899 508 y Fc(\000)944 544 y Fl(\()p Fh(s)p Fl(+)p Fh(t)p Fl(+)p Fh(n)p Fl(\()p Fh(\014)t Fe(\000)p Fh(\013)p Fl(\))p Fe(\000)p Fh(\014)t Fl(\))p Fh(=\014)1264 623 y(n)1313 633 y Ff(1)1671 508 y Fc(\001)1734 478 y(\020)1805 549 y Fh(a)1848 519 y Ff(2)p 1805 565 83 4 v 1809 623 a Fh(c)1845 603 y Ff(2)1899 478 y Fc(\021)1959 498 y Fh(n)2030 478 y Fc(\020)2101 549 y Fh(b)2136 519 y Ff(2)p 2101 565 75 4 v 2101 623 a Fh(c)2137 603 y Ff(2)2187 478 y Fc(\021)2247 498 y Fl(\()p Fh(s)p Fl(+)p Fh(t)p Fe(\000)p Fh(\013n)p Fl(\))p Fh(=\014)2796 588 y Fm(if)33 b Fi(s)27 b Fj(2)h Fm([)p Fi(\013;)17 b(\014)5 b Fm(\),)3495 466 y(\(5)p Fi(:)p Fm(14\))150 827 y(and)34 b(let)f Fi(\016)532 842 y Fh(p)611 827 y Fm(b)s(e)h(the)f(p)s(erio)s(dic)h Fi(\016)t Fm(-function)1492 1064 y Fi(\016)1536 1079 y Fh(p)1583 1064 y Fm(\()p Fi(x)p Fm(\))27 b(=)1853 970 y Fc(X)1850 1184 y Fh(n)p Fe(2)p Fa(Z)2016 1064 y Fi(\016)t Fm(\()p Fi(x)22 b Fj(\000)g Fi(n)p Fm(\))p Fi(:)1087 b Fm(\(5)p Fi(:)p Fm(15\))150 1378 y(Note)30 b(that)g Fi(f)11 b Fm(\()p Fi(s;)17 b(t;)g(n)p Fm(\))28 b(is)i(w)m(ell-de\014ned)i(ev)m (en)g(when)f(\()p Fi(s)16 b Fm(+)h Fi(t)g Fm(+)g Fi(n)p Fm(\()p Fi(\014)22 b Fj(\000)17 b Fi(\013)p Fm(\)\))p Fi(=\014)35 b Fm(is)c(not)f(an)h(in)m(teger.)150 1498 y(F)-8 b(or)33 b Fi(t)g Fm(large,)f Fi(f)11 b Fm(\()p Fi(s;)17 b(t;)g(n)p Fm(\))31 b(is)i(a)g(slo)m(wly-v)-6 b(arying)32 b(function)i(of)f Fi(s)g Fm(and)h Fi(n)p Fm(.)349 1647 y(W)-8 b(e)34 b(then)g(compute,)e(exactly)-8 b(,)667 1967 y Fi(F)731 1982 y Fh(I)777 1967 y Fm(\()p Fi(t)p Fm(\))27 b(=)1023 1872 y Fc(X)1183 1967 y Fi(s)h Fj(2)g Fi(I)7 b(F)1467 1982 y Fh(s)1510 1967 y Fm(\()p Fi(t)p Fm(\))27 b(=)1756 1834 y Fl([)p Fh(t=\013)p Fl(])1768 1872 y Fc(X)1765 2083 y Fh(n)p Fl(=0)1941 1831 y Fc(Z)1996 2058 y Fh(I)2058 1967 y Fi(f)11 b Fm(\()p Fi(s;)17 b(t;)g(n)p Fm(\))p Fi(\016)2473 1982 y Fh(p)2517 1967 y Fm(\(\()p Fi(s)k Fm(+)i Fi(t)f Fj(\000)g Fi(n\013)p Fm(\))p Fi(=\014)5 b Fm(\))262 b(\(5)p Fi(:)p Fm(16\))150 2272 y(Next)28 b(w)m(e)i(appro)m(ximate,)e(b)m(y)h(replacing)g Fi(f)11 b Fm(\()p Fi(s;)17 b(t;)g(n)p Fm(\))26 b(b)m(y)j Fi(f)11 b Fm(\()p Fi(s)2365 2287 y Fl(0)2409 2272 y Fi(;)17 b(t;)g(n)p Fm(\),)28 b(where)h Fi(s)3020 2287 y Fl(0)3094 2272 y Fm(is)g(the)g(midp)s(oin)m(t)150 2392 y(of)35 b Fi(I)8 b Fm(.)48 b(This)36 b(in)m(tro)s(duces)g(an)f(error)f(that)h(is)g(a)f (fraction)h Fi(O)s Fm(\(\001\))f(of)i(the)f(total.)47 b(Since)36 b(the)f(total)150 2511 y(will)d(turn)i(out)f(to)g(b)s(e)g Fi(O)s Fm(\(\001\),)f(the)i(error)f(in)m(tro)s(duced)i(is)e Fi(O)s Fm(\(\001)2510 2475 y Fl(2)2554 2511 y Fm(\).)44 b(W)-8 b(e)34 b(th)m(us)g(ha)m(v)m(e)786 2836 y Fi(F)850 2851 y Fh(I)896 2836 y Fm(\()p Fi(t)p Fm(\))27 b(=)1142 2703 y Fl([)p Fh(t=\013)p Fl(])1154 2742 y Fc(X)1151 2953 y Fh(n)p Fl(=0)1327 2836 y Fi(f)11 b Fm(\()p Fi(s)1473 2851 y Fl(0)1516 2836 y Fi(;)17 b(t;)g(n)p Fm(\))1758 2701 y Fc(Z)1811 2927 y Fh(I)1874 2836 y Fi(\016)1918 2851 y Fh(p)1964 2836 y Fm(\(\()p Fi(s)k Fm(+)i Fi(t)f Fj(\000)h Fi(n\013)p Fm(\))p Fi(=\014)5 b Fm(\))22 b(+)g Fi(O)s Fm(\(\001)3003 2795 y Fl(2)3047 2836 y Fm(\))p Fi(:)381 b Fm(\(5)p Fi(:)p Fm(17\))349 3166 y(Next)48 b(w)m(e)h(use)h(the)f(fact)g(that)f(m)m(ultiples)g(of)h(an)g (irrational)e(n)m(um)m(b)s(er)i(are)g(uniformly)150 3286 y(distributed)41 b(on)f Fb(R)5 b Fi(=)p Fb(Z)p Fm(.)68 b(F)-8 b(or)40 b(an)m(y)g(desired)h(degree)g(of)f(accuracy)-8 b(,)42 b(one)f(can)f(\014nd)h(an)f Fi(N)51 b Fm(suc)m(h)150 3405 y(that,)31 b(for)g(an)m(y)g Fi(N)42 b Fm(consecutiv)m(e)32 b(in)m(tegers)f Fi(n)1800 3420 y Fh(i)1833 3405 y Fm(,)1892 3325 y Fc(R)1939 3441 y Fh(I)2002 3405 y Fi(\016)2046 3420 y Fh(p)2092 3405 y Fm(\(\()p Fi(s)17 b Fm(+)h Fi(t)f Fj(\000)h Fi(n)2537 3420 y Fh(i)2570 3405 y Fi(\013)p Fm(\))p Fi(=\014)5 b Fm(\))31 b(equals)g(one)g(a)g(fraction)150 3525 y(\001)p Fi(=\014)46 b Fm(of)41 b(the)h(time)d(\(to)h(within)h (the)g(allo)m(w)m(ed)g(error\),)h(and)f(0)g(the)g(rest)g(of)h(the)f (time.)65 b(If)42 b(w)m(e)150 3644 y(restrict)h(ourselv)m(es)h(to)e Fi(t)h Fm(so)g(large)f(that)h Fi(f)11 b Fm(\()p Fi(s)1903 3659 y Fl(0)1946 3644 y Fi(;)17 b(t;)g(n)p Fm(\))41 b(is)i(nearly)g (constan)m(t)g(as)g Fi(n)g Fm(v)-6 b(aries)43 b(o)m(v)m(er)150 3764 y(in)m(terv)-6 b(als)33 b(of)h(size)g Fi(N)11 b Fm(,)32 b(then)530 3950 y Fl([)p Fh(t=\013)p Fl(])542 3989 y Fc(X)539 4200 y Fh(n)p Fl(=0)715 4083 y Fi(f)11 b Fm(\()p Fi(s)861 4098 y Fl(0)905 4083 y Fi(;)17 b(t;)g(n)p Fm(\))1147 3948 y Fc(Z)1200 4174 y Fh(I)1262 4083 y Fi(\016)1306 4098 y Fh(p)1352 4083 y Fm(\(\()p Fi(s)22 b Fm(+)g Fi(t)g Fj(\000)h Fi(n\013)p Fm(\))p Fi(=\014)5 b Fm(\))27 b Fj(\031)i Fm(\(\001)p Fi(=\014)5 b Fm(\))2491 3950 y Fl([)p Fh(t=\013)p Fl(])2502 3989 y Fc(X)2499 4200 y Fh(n)p Fl(=0)2675 4083 y Fi(f)11 b Fm(\()p Fi(s)2821 4098 y Fl(0)2864 4083 y Fi(;)17 b(t;)g(n)p Fm(\))p Fi(:)378 b Fm(\(5)p Fi(:)p Fm(18\))150 4383 y(Finally)-8 b(,)33 b(the)i Fi(t)30 b Fj(!)g(1)k Fm(limit)e(of)j(the)g(resulting)g(sum)f (can)h(b)s(e)g(expressed)i(as)d(an)h(in)m(tegral,)f(and)150 4503 y(yields)f(a)g(nonzero)h(n)m(um)m(b)s(er.)p 1346 4513 60 80 v 349 4652 a(W)-8 b(e)39 b(no)m(w)g(turn,)i(as)d(in)h Fj(x)p Fm(4,)h(to)e(the)h(join)m(t)g(distribution)g(of)g(sizes)g(and)h (orien)m(tation.)60 b(W)-8 b(e)150 4772 y(parametrize)26 b(the)i(space)g(of)f(p)s(ossible)g(orien)m(tations)g(of)g(a)g(single)g (size,)h(as)f(in)g Fj(x)q Fm(4,)g(b)m(y)g(t)m(w)m(o)g(copies)150 4891 y(of)34 b(the)f(unit)h(circle,)f(with)g(Haar)g(measure)h Fi(d)p Fm(\012.)44 b(\(See)34 b(Figure)g(8\).)44 b(The)34 b(space)g(of)g(all)e(p)s(ossible)150 5011 y(tiles)h(up)h(to)f (translation,)f(whic)m(h)i(w)m(e)f(denote)h Fi(X)8 b Fm(,)33 b(is)g(2)g(copies)h(of)f Fi(S)2720 4975 y Fl(1)2787 5011 y Fj(\002)23 b Fm([0)p Fi(;)17 b(\026)p Fm(\).)349 5160 y(Let)40 b Fi(f)49 b Fm(b)s(e)40 b(a)f(function)h(on)f Fi(X)8 b Fm(.)61 b(Giv)m(en)39 b(a)g(collection)g Fi(S)45 b Fm(of)39 b(tiles,)h(let)f Fi(<)f(f)5 b(;)17 b(S)43 b(>)c Fm(b)s(e)h(the)150 5280 y(a)m(v)m(erage)29 b(v)-6 b(alue)29 b(of)g Fi(f)39 b Fm(on)29 b(the)g(individual)f(tiles)h(in)g Fi(S)6 b Fm(,)29 b(where)g(eac)m(h)h(tile)e(is)h(giv)m(en)g(equal)f(w)m (eigh)m(t.)1900 5519 y(26)p eop %%Page: 27 27 27 26 bop 150 100 a Fm(Let)32 b Fi(<)c(f)5 b(;)17 b(S)33 b(>)702 63 y Fe(0)762 100 y Fm(b)s(e)f(the)g(a)m(v)m(erage)f(v)-6 b(alue)32 b(of)g Fi(f)42 b Fm(on)32 b(the)f(individual)g(tiles)h(in)f Fi(S)6 b Fm(,)32 b(where)g(eac)m(h)h(tile)150 219 y(is)g(w)m(eigh)m (ted)g(prop)s(ortionally)e(to)i(its)f(area.)44 b(Let)33 b(\()p Fi(f)5 b(;)17 b(S)6 b Fm(\))31 b(b)s(e)i(the)g(sum)f(of)h Fi(f)44 b Fm(on)32 b(the)i(individual)150 339 y(tiles)f(of)g Fi(S)6 b Fm(.)44 b(W)-8 b(e)34 b(de\014ne)h(a)e(semigroup)f Fi(e)1678 303 y Fh(tL)1763 273 y Fd(\003)1843 339 y Fm(acting)h(on)h (functions)g(on)f Fi(X)41 b Fm(b)m(y)1431 574 y(\()p Fi(e)1516 533 y Fh(tL)1601 502 y Fd(\003)1648 574 y Fi(f)5 b(;)17 b(S)6 b Fm(\))26 b(=)j(\()p Fi(f)5 b(;)17 b(e)2169 533 y Fh(tL)2258 574 y Fm(\()p Fi(S)6 b Fm(\)\))p Fi(:)1025 b Fm(\(5)p Fi(:)p Fm(19\))150 839 y Fn(Theorem)51 b(9)45 b Fk(Assume)i(a)f(tiling)g Fi(T)14 b(il)r Fm(\()p Fi(z)t Fm(\))p Fk(,)49 b(with)d Fi(z)51 b Fk(irr)-5 b(ational)46 b(and)g(with)g Fi(\022)k Fm(=)d(tan)3420 802 y Fe(\000)p Fl(1)3527 839 y Fm(\()p Fi(a=b)p Fm(\))150 958 y Fk(an)e(irr)-5 b(ational)44 b(multiple)j(of)d Fi(\031)t Fk(.)73 b(L)-5 b(et)45 b Fi(f)56 b Fk(b)-5 b(e)44 b(a)h(c)-5 b(ontinuous)45 b(function)g(on)g Fi(X)8 b Fk(,)46 b(and)f(let)g Fj(f)p Fi(S)3646 973 y Fh(n)3700 958 y Fj(g)150 1078 y Fk(b)-5 b(e)41 b(a)g(se)-5 b(quenc)g(e)40 b(of)g(sup)-5 b(ertiles)41 b(of)f(incr)-5 b(e)g(asing)39 b(size.)61 b(L)-5 b(et)41 b Fi(d\027)j Fm(=)38 b Fi(e)2678 1041 y Fl(2)p Fh(s)2760 1078 y Fm(\()p Fi(a)2852 1041 y Fl(2)2896 1078 y Fi(h)p Fm(\()p Fi(\013)27 b Fj(\000)f Fi(s)p Fm(\))g(+)g Fi(b)3444 1041 y Fl(2)3489 1078 y Fi(h)p Fm(\()p Fi(\014)32 b Fj(\000)150 1197 y Fi(s)p Fm(\)\))p Fi(d)p Fm(\012)p Fi(ds=)p Fm(2)p Fi(c)641 1161 y Fl(2)720 1197 y Fk(and)j Fi(d\032)27 b Fm(=)h(2)p Fi(c)1242 1161 y Fl(2)1287 1197 y Fi(e)1333 1161 y Fe(\000)p Fl(2)p Fh(s)1478 1197 y Fi(d\027)6 b(=)p Fm(\()p Fi(a)1727 1161 y Fl(2)1771 1197 y Fi(\013)22 b Fm(+)h Fi(b)2000 1161 y Fl(2)2045 1197 y Fi(\014)5 b Fm(\))36 b Fk(b)-5 b(e)35 b(me)-5 b(asur)g(es)35 b(on)h Fi(X)8 b Fk(.)44 b(Then)1445 1473 y Fm(lim)1410 1533 y Fh(n)p Fe(!1)1646 1473 y Fi(<)28 b(f)5 b(;)17 b(S)1911 1488 y Fh(n)1992 1473 y Fi(>)p Fm(=)2175 1338 y Fc(Z)2230 1564 y Fh(X)2323 1473 y Fi(f)11 b(d\027)959 b Fm(\(5)p Fi(:)p Fm(20)p Fi(a)p Fm(\))1419 1815 y(lim)1385 1875 y Fh(n)p Fe(!1)1620 1815 y Fi(<)28 b(f)5 b(;)17 b(S)1885 1830 y Fh(n)1967 1815 y Fi(>)2044 1774 y Fe(0)2072 1815 y Fm(=)2177 1679 y Fc(Z)2233 1906 y Fh(X)2325 1815 y Fi(f)11 b(d\032:)936 b Fm(\(5)p Fi(:)p Fm(20)p Fi(b)p Fm(\))150 2043 y Fk(These)35 b(limits)h(also)f(apply)i(if)d Fi(f)47 b Fk(is)34 b(the)i(char)-5 b(acteristic)34 b(function)h(of)g(a) g(r)-5 b(e)g(ctangle)36 b(in)f Fi(X)8 b Fk(.)150 2192 y(Pr)-5 b(o)g(of:)48 b Fm(The)36 b(pro)s(of)g(is)g(extremely)e(similar) g(to)h(the)i(pro)s(of)f(of)g(Theorem)f(7.)52 b(As)35 b(in)h(that)g(case,)150 2312 y(it)i(is)h(su\016cien)m(t)h(to)f (establish)g(\(5.20a\))f(for)h(an)g(arbitrary)f(con)m(tin)m(uous)i (function)g Fi(f)11 b Fm(.)61 b(Suc)m(h)40 b(a)150 2431 y(function)j(can)f(b)s(e)f(written)h(as)f(an)h(absolutely)f(con)m(v)m (ergen)m(t)i(sum)e(of)h(F)-8 b(ourier)41 b(mo)s(des)h(\(with)150 2551 y(resp)s(ect)h(to)e(rotations\).)69 b(The)43 b(co)s(e\016cien)m(t) f(of)g(eac)m(h)h(mo)s(de)e(is)h(a)g Fb(C)2741 2515 y Fl(2)2833 2551 y Fm(v)-6 b(alued)43 b(function)g(of)f Fi(s)p Fm(.)150 2670 y(The)34 b(op)s(erator)g Fi(e)804 2634 y Fh(tL)889 2604 y Fd(\003)969 2670 y Fm(comm)m(utes)f(with)h (rotation,)e(and)j(so)f(acts)g(separately)f(on)h(eac)m(h)h(F)-8 b(ourier)150 2790 y(mo)s(de.)349 2939 y(On)34 b(the)g Fi(n)p Fm(-th)f(F)-8 b(ourier)33 b(mo)s(de,)f Fi(e)1594 2903 y Fh(tL)1679 2873 y Fd(\003)1726 2939 y Fm(,)h(for)h Fi(t)f Fm(small,)e(acts)i(as)h(follo)m(ws.)325 3108 y Fc(\020)385 3219 y Fi(e)431 3178 y Fh(tL)516 3147 y Fd(\003)562 3219 y Fi(f)622 3108 y Fc(\021)698 3219 y Fm(\()p Fi(s)p Fm(\))27 b(=)955 3078 y Fc(\032)1046 3159 y Fi(A )t Fm(\()p Fi(s)21 b Fj(\000)i Fi(t)f Fm(+)g Fi(\013)p Fm(\))g(+)h Fi(B)5 b( )t Fm(\()p Fi(s)21 b Fj(\000)h Fi(t)g Fm(+)h Fi(\014)5 b Fm(\))100 b(if)33 b Fi(s)28 b Fj(2)g Fm([0)p Fi(;)17 b(t)p Fm(\);)1046 3278 y Fi( )t Fm(\()p Fi(s)k Fj(\000)i Fi(t)p Fm(\))1096 b(all)32 b(other)i Fi(s)27 b Fj(2)h Fm([0)p Fi(;)17 b(\026)p Fm(\),)3495 3219 y(\(5)p Fi(:)p Fm(21\))150 3498 y(where)34 b(the)g(matrices)e Fi(A)h Fm(and)h Fi(B)j Fm(are,)c(as)h(in)f Fj(x)p Fm(4,)935 3777 y Fi(A)28 b Fm(=)1143 3637 y Fc(\022)1563 3717 y Fm(0)430 b(exp[)p Fj(\000)p Fi(in)p Fm(\()p Fi(\022)25 b Fm(+)d Fi(\031)t(=)p Fm(2\)])1233 3837 y(exp[)p Fi(in)p Fm(\()p Fi(\022)i Fm(+)f Fi(\031)t(=)p Fm(2\)])467 b(0)2847 3637 y Fc(\023)2937 3777 y Fi(;)530 b Fm(\(5)p Fi(:)p Fm(22\))349 4132 y Fi(B)32 b Fm(=)562 3991 y Fc(\022)652 4072 y Fm(exp\()p Fi(in\022)s Fm(\))21 b(+)i(exp\()p Fi(in)p Fm(\()p Fi(\022)h Fm(+)f Fi(\031)t Fm(\)\))481 b(2)17 b(exp\()p Fj(\000)p Fi(in\022)s Fm(\))996 4191 y(2)g(exp\()p Fi(in\022)s Fm(\))443 b(exp\()p Fj(\000)p Fi(in\022)s Fm(\))22 b(+)g(exp\()p Fi(in)p Fm(\()p Fj(\000)p Fi(\022)j Fm(+)e Fi(\031)t Fm(\)\))3179 3991 y Fc(\023)3269 4132 y Fi(:)198 b Fm(\(5)p Fi(:)p Fm(23\))349 4413 y(The)50 b(0-th)g(F)-8 b(ourier)50 b(mo)s(de)f(decouples)i(in)m(to)2127 4333 y Fc(\000)2173 4369 y Fl(1)2173 4448 y(1)2213 4333 y Fc(\001)2308 4413 y Fm(and)2518 4333 y Fc(\000)2595 4369 y Fl(1)2564 4448 y Fe(\000)p Fl(1)2666 4333 y Fc(\001)2761 4413 y Fm(comp)s(onen)m(ts.)94 b(The)3619 4333 y Fc(\000)3665 4369 y Fl(1)3665 4448 y(1)3704 4333 y Fc(\001)150 4533 y Fm(comp)s(onen)m(t)34 b(is)h(the)g(distribution)f(of)h(sizes)g (regardless)g(of)g(orien)m(tation,)f(and)h(its)f(asymptotic)150 4652 y(b)s(eha)m(vior)g(w)m(as)g(already)f(computed)h(in)g(Theorem)f (8.)45 b(W)-8 b(e)34 b(m)m(ust)g(sho)m(w)g(that)g(the)g Fi(n)28 b Fm(=)h(0)3557 4572 y Fc(\000)3633 4608 y Fl(1)3602 4687 y Fe(\000)p Fl(1)3704 4572 y Fc(\001)150 4772 y Fm(comp)s(onen)m(t,)g(and)f(all)f(the)i(F)-8 b(ourier)28 b(mo)s(des)f(with)h Fi(n)g Fj(6)p Fm(=)g(0,)g(gro)m(w)g(strictly)f(slo) m(w)m(er)i(than)f(the)g(size)150 4891 y(of)33 b(the)h(system,)e(and)i (so)f(represen)m(t)i(a)e(decreasing)i(fraction)e(of)g(the)h(system.)349 5041 y(W)-8 b(e)38 b(will)f(con)m(trol)g(the)h Fi(L)1301 5005 y Fl(1)1383 5041 y Fm(norms)f(of)h(the)f(un)m(w)m(an)m(ted)j(F)-8 b(ourier)37 b(mo)s(des.)56 b(T)-8 b(o)37 b(do)h(this)g(w)m(e)150 5160 y(need)44 b(the)f Fi(L)641 5124 y Fl(1)728 5160 y Fm(norms)f(of)h(the)f(matrices)g Fi(A)g Fm(and)h Fi(B)5 b Fm(,)45 b(and)e(v)-6 b(arious)42 b(pro)s(ducts)i(of)e Fi(A)h Fm(and)g Fi(B)5 b Fm(.)150 5280 y(The)32 b Fi(L)421 5244 y Fl(1)496 5280 y Fm(norm)f(of)g(a)g(matrix)e(is)i(maxim)m(um,)d (o)m(v)m(er)j(all)f(columns,)h(of)h(the)f(sum)g(of)g(the)h(absolute) 1900 5519 y(27)p eop %%Page: 28 28 28 27 bop 150 100 a Fm(v)-6 b(alues)33 b(of)g(the)f(en)m(tries)i(in)e (that)g(column.)44 b(One)33 b(can)g(get)f(a)h(b)s(ound)g(on)g(the)g (gro)m(wth)f(of)h(the)f Fi(L)3705 63 y Fl(1)150 219 y Fm(norm)f(of)i(a)f(mo)s(de)g(b)m(y)g(the)h(mo)s(de)e(with)h(its)g (absolute)h(v)-6 b(alue,)32 b(and)h(replacing)f(the)h(matrices)e Fi(A)150 339 y Fm(and)j Fi(B)j Fm(b)m(y)d(their)f(norms.)349 488 y(F)-8 b(or)37 b Fi(n)d Fm(=)h(0,)j Fi(B)935 407 y Fc(\000)1011 444 y Fl(1)980 523 y Fe(\000)p Fl(1)1082 407 y Fc(\001)1162 488 y Fm(=)d(0.)56 b(With)38 b Fi(B)h Fm(=)c(0)i(it)g(is)g(as)g(if)h(there)g(is)f(only)g(one)g(daugh)m(ter)i (p)s(er)150 620 y(division,)30 b(hence)j(the)e Fi(L)1056 584 y Fl(1)1132 620 y Fm(norm)f(of)h(the)1670 539 y Fc(\000)1747 576 y Fl(1)1716 654 y Fe(\000)p Fl(1)1818 539 y Fc(\001)1894 620 y Fm(mo)s(de)g(at)f(time)g Fi(t)h Fm(is)f(b)s(ounded)j(b)m(y)e(the) g Fi(L)3447 584 y Fl(1)3523 620 y Fm(norm)150 740 y(of)38 b(the)g(mo)s(de)f(at)g(time)f(0.)57 b(Hence,)39 b(as)f(a)g(fraction)f (of)h(the)g(system,)g(this)g(mo)s(de)e(shrinks)j(lik)m(e)150 859 y Fi(e)196 823 y Fe(\000)p Fl(2)p Fh(s)341 859 y Fm(.)349 1009 y(Next)e(w)m(e)g(consider)h Fi(n)f Fm(o)s(dd,)h(for)f (whic)m(h)h(the)f(diagonal)f(terms)h(in)g Fi(B)k Fm(v)-6 b(anish.)56 b(The)37 b(sum)150 1128 y(of)f(the)f(absolute)h(v)-6 b(alues)36 b(of)g(the)g(en)m(tries)g(of)g(eac)m(h)g(column)f(of)h Fi(B)k Fm(equals)35 b(t)m(w)m(o.)51 b(This)35 b(is)h(as)f(if,)150 1248 y(at)h(eac)m(h)i(sub)s(division,)g(only)e(t)m(w)m(o)h(daugh)m(ter) h(B)f(tiles)f(are)h(pro)s(duced,)j(instead)d(of)g(4.)55 b(T)-8 b(o)37 b(put)150 1367 y(it)c(another)h(w)m(a)m(y)-8 b(,)33 b(at)g(eac)m(h)i(sub)s(division)e(a)h(fraction)f Fi(b)2201 1331 y Fl(2)2246 1367 y Fi(=)p Fm(2)p Fi(c)2389 1331 y Fl(2)2467 1367 y Fm(of)h(the)g(area)f(is)h(lost.)45 b(Since)34 b(eac)m(h)150 1487 y(piece)e(of)g(a)g(tile)f(of)h(size)g Fj(\000)p Fi(t)g Fm(m)m(ust)f(b)s(e)h(divided)g(at)f(least)h Fi(t=\014)37 b Fm(times,)30 b(this)i(means)f(that)h(the)f Fi(L)3705 1451 y Fl(1)150 1606 y Fm(norm)g(of)i(the)f Fi(n)p Fm(-th)g(mo)s(de,)f(for)i Fi(n)f Fm(o)s(dd,)g(can)h(gro)m(w)e (no)i(faster)f(than)h Fi(e)2783 1570 y Fl(2)p Fh(t)2858 1606 y Fm([1)19 b Fj(\000)i Fm(\()p Fi(b)3135 1570 y Fl(2)3179 1606 y Fi(=)p Fm(2)p Fi(c)3322 1570 y Fl(2)3366 1606 y Fm(\)])3433 1570 y Fh(t=\014)3589 1606 y Fm(and)150 1726 y(so,)33 b(as)g(a)g(fraction)h(of)f(the)h(system,)e(go)s(es)h(to)g (zero.)349 1875 y(Finally)c(w)m(e)g(consider)i Fi(n)e Fm(ev)m(en)h(but)g(nonzero.)44 b(Here)30 b(the)g(column)f(sums)g(of)h Fi(A)f Fm(and)h Fi(B)k Fm(are)150 1995 y(the)29 b(same)f(as)g(in)h(the) f Fi(n)g Fm(=)g(0)g(case,)i(namely)d(1)h(and)h(4,)g(resp)s(ectiv)m(ely) -8 b(.)42 b(Ho)m(w)m(ev)m(er,)30 b(the)f Fi(L)3411 1959 y Fl(1)3483 1995 y Fm(norms)150 2114 y(of)36 b(v)-6 b(arious)35 b(pro)s(ducts)h(of)g Fi(A)f Fm(and)h Fi(B)k Fm(are)35 b(smaller)f(that)h(in)g(the)h Fi(n)31 b Fm(=)h(0)j(mo)s(de.)49 b(F)-8 b(or)35 b(example,)150 2234 y Fi(B)231 2198 y Fl(2)303 2234 y Fm(=)28 b(4)17 b(cos\()p Fi(n\022)s Fm(\))p Fi(B)31 b Fm(has)d(norm)f(16)p Fj(j)17 b Fm(cos)o(\()p Fi(n\022)s Fm(\))p Fj(j)p Fm(,)27 b(whic)m(h)i(is)f(strictly)e(smaller) h(than)h(16.)42 b(The)28 b(norm)150 2353 y(of)e Fi(B)5 b(AB)30 b Fm(is)c(also)g(16)p Fj(j)17 b Fm(cos)o(\()p Fi(n\022)s Fm(\))p Fj(j)p Fm(.)40 b(Indeed,)29 b(the)d(only)g(w)m(ords) g(in)g Fi(A)g Fm(and)g Fi(B)31 b Fm(whic)m(h)c(ha)m(v)m(e)f(norms)g(as) 150 2473 y(large)k(as)g(in)h(the)f Fi(n)e Fm(=)g(0)i(case)h(are)g Fi(A)1512 2437 y Fh(m)1617 2473 y Fm(and)g Fi(A)1883 2437 y Fh(m)1954 2447 y Ff(1)1998 2473 y Fi(B)5 b(A)2154 2437 y Fh(m)2225 2447 y Ff(2)2267 2473 y Fm(.)43 b(Since)32 b(the)e(expansion)h(of)g Fi(e)3381 2437 y Fh(tL)3466 2407 y Fd(\003)3513 2473 y Fi(f)11 b Fm(,)30 b(for)150 2593 y Fi(t)h(>)h(\014)5 b Fm(,)36 b(in)m(v)m(olv)m(es)f(expressions)i (suc)m(h)g(as)e Fi(B)1781 2556 y Fl(2)1826 2593 y Fm(,)g(the)h(gro)m (wth)f(of)h(the)g Fi(L)2758 2556 y Fl(1)2837 2593 y Fm(norm)f(of)h Fi(n)p Fm(-th)f(F)-8 b(ourier)150 2712 y(mo)s(de)34 b(is)h(b)s(ounded)i (b)m(y)e(an)g(exp)s(onen)m(t)h(strictly)e(less)i(than)f(2.)49 b(As)35 b(a)g(fraction)g(of)h(the)f(system,)150 2832 y(the)f Fi(n)p Fm(-th)f(mo)s(de)f(go)s(es)i(to)e(zero.)p 1460 2842 60 80 v 150 2981 a Fn(Remark:)71 b Fm(The)48 b(sp)s(ectrum)e(of)h Fi(e)1490 2945 y Fh(tL)1575 2915 y Fd(\003)1669 2981 y Fm(ma)m(y)e(b)s(e)i(obtained)g(exactly)f(as)h(in) f Fj(x)q Fm(4.)84 b(In)47 b(seeking)150 3101 y(eigen)m(v)m(ectors,)34 b(equation)f(\(4.20\))e(is)j(replaced)g(b)m(y)1185 3488 y(\()p Fi(e)1270 3447 y Fh(\026\025)1393 3488 y Fj(\000)23 b Fi(e)1539 3447 y Fl(\()p Fh(\026)p Fe(\000)p Fh(\014)t Fl(\))p Fh(\025)1813 3488 y Fi(B)k Fj(\000)c Fi(e)2062 3447 y Fl(\()p Fh(\026)p Fe(\000)p Fh(\013)p Fl(\))p Fh(\025)2339 3488 y Fi(A)p Fm(\))p Fi(v)31 b Fm(=)d(0)p Fi(:)780 b Fm(\(5)p Fi(:)p Fm(24\))150 3875 y(Subsequen)m(t)39 b(analysis)d(ma)m(y)g(b)s(e)h(rep)s(eated)h(w)m(ord)f(for)g(w)m(ord,)h (replacing)f Fi(\025)g Fm(b)m(y)g Fi(e)3226 3839 y Fh(\025)3278 3875 y Fm(,)h Fi(p)e Fm(b)m(y)h Fi(\013)p Fm(,)h Fi(q)150 3995 y Fm(b)m(y)33 b Fi(\014)5 b Fm(,)34 b Fi(m)g Fm(b)m(y)f Fi(\026)p Fm(,)g Fi( )856 4010 y Fh(k)938 3995 y Fm(b)m(y)g Fi( )t Fm(\()p Fi(s)p Fm(\),)f(and)i Fi(r)1572 3959 y Fh(k)1654 3995 y Fm(b)m(y)f Fi(e)1838 3959 y Fe(\000)p Fh(s)1943 3995 y Fm(.)150 4144 y Fj(x)p Fn(6.)51 b(Tw)m(o)38 b(exceptional)e(tilings)h(|)h Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(3\))37 b Fn(and)h Fi(T)14 b(il)r Fm(\(2\))349 4294 y(W)-8 b(e)28 b(sa)m(w)f(in)g Fj(x)q Fm(4)g(ho)m(w)g(the)g(p)s (opulation)g(statistics)g(of)g(a)g(rational)f(tiling)g Fi(T)14 b(il)r Fm(\()p Fi(p=q)t Fm(\))27 b(dep)s(ends)150 4413 y(on)j Fi(p)h Fm(and)f Fi(q)t Fm(.)43 b(If)31 b Fi(q)g(>)e Fm(1,)h(the)g(second)i(eigen)m(v)-6 b(alue)31 b(of)f(the)h(p)s(opulation)f(matrix)e(is)i(greater)g(than)150 4533 y(one,)51 b(and)d(\015uctuations)g(increase)g(with)f(sub)s (division.)86 b(This)48 b(leads)g(to)f(phenomena)g(suc)m(h)150 4652 y(as)42 b(slippage)h(along)f(fault)g(lines)h(and)f(a)g(failure)h (to)f(b)s(e)g(globally)f(edge-to-edge.)73 b(A)42 b(t)m(ypical)150 4772 y(example,)32 b Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(2\),)32 b(w)m(as)h(studied)i(in)e Fj(x)p Fm(3.)349 4921 y(In)i(this)g(section)h (w)m(e)f(study)g(examples)g(of)g(the)g(remaining)f(cases.)50 b(W)-8 b(e)35 b(study)h Fi(T)14 b(il)r Fm(\(2\))34 b(as)150 5041 y(an)42 b(example)f(of)h(a)g Fi(T)14 b(il)r Fm(\()p Fi(p=)p Fm(1\))41 b(tiling.)69 b(In)43 b(all)e(suc)m(h)i(tilings,)g (the)f(second)i(eigen)m(v)-6 b(alue)42 b(of)h(the)150 5160 y(p)s(opulation)e(matrix)e(is)i(less)g(than)g(one.)68 b(W)-8 b(e)42 b(shall)f(see)g(ho)m(w,)i(in)f(the)f(case)h(of)f Fi(T)14 b(il)r Fm(\(2\),)42 b(this)150 5280 y(prev)m(en)m(ts)35 b(slippage)e(along)g(fault)g(lines.)1900 5519 y(28)p eop %%Page: 29 29 29 28 bop 1039 2050 a @beginspecial 50 @llx 20 @lly 410 @urx 430 @ury 1800 @rwi @setspecial %%BeginDocument: fig10.eps /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def end gnudict begin gsave 50 320 translate 0.050 0.050 scale 0 setgray /Helvetica findfont 300 scalefont setfont newpath LTa LTb LT0 6486 6305 M %('fig10.out1') Rshow %6570 6305 M %252 0 V 672 211 M 3561 913 L 3732 211 L 672 211 L 6450 1615 M 3561 913 L 3732 211 L 6450 1615 L 3732 211 M 6621 913 L -171 702 V 3732 211 L 6621 913 L 6791 211 L -3059 0 V 3059 0 R 6450 1615 L 341 83 V 0 -1487 V 3400 400 M (1) show 5000 1000 M (2) show 5300 700 M (3) show 6300 1000 M (4) show 6500 400 M (5) show 3000 -50 M (Figure 10a) show stroke grestore gsave 50 185 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V 0 -351 V 80 331 V -80 20 V 0 -351 V 3561 562 M 3481 231 L 80 -20 V 0 351 V -80 331 V 80 20 V 0 -351 V 5167 952 M -81 331 V -80 -19 V 5167 952 L -224 257 V 63 55 V 5167 952 L 3722 601 M 3946 344 L -63 -55 V 3722 601 L -80 331 V -81 -19 V 3722 601 L 5015 874 M 81 -332 V 80 20 V 5015 874 L 5239 616 L -63 -54 V 5015 874 L 1445 351 R -224 257 V 63 54 V 161 -311 V 80 -332 V 81 20 V -161 312 V 5176 211 M -80 331 V 80 20 V 0 -351 V 81 331 V -81 20 V 0 -351 V 6621 562 M 6540 231 L 81 -20 V 0 351 V -81 331 V 81 20 V 0 -351 V 3000 -50 M (Figure 10c) show stroke grestore end showpage %%EndDocument @endspecial 876 2230 a Fm(Figure)33 b(10.)44 b(Three)34 b(stages)f(of)h(sub)s(division)f(for)g Fi(T)14 b(il)r Fm(\(2\).)349 2439 y(Finally)-8 b(,)40 b(w)m(e)g(consider)g Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(3\),)40 b(the)g(only)f(rational)g (tiling)g(to)g(exhibit)g(only)g(a)h(\014nite)150 2559 y(n)m(um)m(b)s(er)45 b(of)g(orien)m(tations)g(of)f(eac)m(h)i(size)f(of) g(tile.)78 b(As)44 b(in)h(all)f(cases)h(with)g Fi(q)51 b(>)c Fm(1,)g(there)e(is)150 2678 y(an)37 b(eigen)m(v)-6 b(alue)38 b(greater)f(than)g(one)g(in)g(the)h(problem.)55 b(The)37 b(\015uctuations)h(go)m(v)m(erned)g(b)m(y)f(this)150 2798 y(eigen)m(v)-6 b(alue)44 b(are)g(enough)g(to)f(force)i Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(3\))42 b(to)h(b)s(e)h(nonp)s(erio)s (dic.)76 b(Ho)m(w)m(ev)m(er,)46 b(b)s(ecause)f(of)150 2917 y(rational)29 b(relations)h(b)s(et)m(w)m(een)h(the)g(lengths)g(of) f(certain)g(edges,)i(the)e(tiles)g(in)g Fi(T)14 b(il)r Fm(\(1)p Fi(=)p Fm(3\))29 b(meet)h(in)150 3037 y(only)j(a)g(\014nite)g (n)m(um)m(b)s(er)h(of)g(w)m(a)m(ys.)349 3186 y(W)-8 b(e)33 b(b)s(egin)f(with)g Fi(T)14 b(il)r Fm(\(2\).)43 b Fi(T)14 b(il)r Fm(\(2\))31 b(is)h(based)h(on)f(the)h(righ)m(t)f(triangle)f (with)h Fi(a)27 b Fm(=)3345 3104 y Fj(p)p 3428 3104 50 4 v 82 x Fm(5)20 b Fj(\000)g Fm(2)28 b Fj(\031)150 3318 y Fm(0)p Fi(:)p Fm(2361,)40 b Fi(b)f Fm(=)g(2)744 3219 y Fc(p)p 844 3219 305 4 v 16 x Fj(p)p 927 3235 50 4 v 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Fe(\006)1295 1252 y Fm(,)g(it)g(is)g (easy)g(to)g(see)h(that)259 1478 y(\()298 1391 y Fj(p)p 381 1391 V 87 x Fm(5)22 b Fj(\000)g Fm(2\))p Fi(H)724 1493 y Fh(n)800 1478 y Fj(\000)h Fi(S)961 1493 y Fh(n)1043 1478 y Fm(=)28 b Fj(\000)p Fm(\(2)22 b Fj(\000)1436 1391 y(p)p 1519 1391 V 87 x Fm(5\))1608 1437 y Fh(n)p Fl(+1)1762 1478 y Fm(;)216 b(\()2045 1391 y Fj(p)p 2128 1391 V 87 x Fm(5)22 b Fj(\000)g Fm(2\))p Fi(H)2479 1437 y Fe(0)2471 1503 y Fh(n)2547 1478 y Fj(\000)h Fi(S)2714 1437 y Fe(0)2708 1503 y Fh(n)2790 1478 y Fm(=)28 b Fj(\000)p Fm(\(2)22 b Fj(\000)3183 1391 y(p)p 3266 1391 V 87 x Fm(5)o(\))3354 1437 y Fh(n)3408 1478 y Fi(:)109 b Fm(\(6)p Fi(:)p Fm(2\))349 1723 y(Next)34 b(w)m(e)g(measure)g(slippage.)47 b(Let)34 b Fi(P)48 b Fm(and)34 b Fi(R)g Fm(b)s(e)h(the)f(endp)s(oin)m(ts)h(of)g (a)e(h)m(yp)s(oten)m(use,)j(as)150 1843 y(in)30 b(Figure)h(7,)f(and)h (let)f Fi(E)36 b Fm(b)s(e)30 b(an)m(y)h(in)m(termediate)e(p)s(oin)m(t,) i(not)f(necessarily)h(the)g(midp)s(oin)m(t.)42 b(Let)150 1962 y Fi(f)199 1977 y Fh(n)253 1962 y Fm(\()p Fi(E)6 b Fm(\))33 b(b)s(e)i(the)f(n)m(um)m(b)s(er)h(of)f(complete)g(short)g (legs,)h(b)s(et)m(w)m(een)g Fi(P)48 b Fm(and)35 b Fi(E)6 b Fm(,)33 b(in)i Fi(\033)3145 1926 y Fh(n)3198 1962 y Fm(\()p Fi(P)14 b(R)q Fm(\),)33 b(min)m(us)150 2082 y(the)f(n)m(um)m(b) s(er)f(of)h(short)f(legs)h(b)s(et)m(w)m(een)g Fi(P)45 b Fm(and)32 b Fi(E)k Fm(in)c Fi(\033)2207 2046 y Fh(n)2260 2082 y Fm(\()p Fi(R)q(P)14 b Fm(\).)42 b(As)32 b(in)f(the)g(pro)s(of)h (of)f(Theorem)150 2202 y(5,)40 b Fi(f)317 2217 y Fh(n)371 2202 y Fm(\()p Fi(E)6 b Fm(\))37 b(measures)i(the)g(exten)m(t)g(to)f (whic)m(h)i(the)f(t)m(w)m(o)f(tiles)h(of)g Fi(T)2668 2217 y Fl(2)p Fh(n)p Fl(+2)2901 2202 y Fm(that)g(meet)f(at)g Fi(E)44 b Fm(are)150 2321 y(o\013set.)150 2471 y Fn(Lemma)38 b(10)c Fj(j)p Fi(f)781 2486 y Fh(n)835 2471 y Fm(\()p Fi(E)6 b Fm(\))p Fj(j)26 b Fi(<)i Fm(5.)150 2620 y Fk(Pr)-5 b(o)g(of:)57 b Fm(Let)41 b Fi(v)705 2635 y Fl(0)790 2620 y Fm(=)f Fi(P)14 b Fm(,)42 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