2:00 pm Thursday, April 10, 2014
Algebra, Number Theory, and Combinatorics : Some arithmetic questions related to Sierpinski's triangle by Eduardo Duenez (University of Texas at San Antonio) in RLM 9.166
The Sierpinski triangle S supports a natural invariant probability measure mu. When S is projected on any fixed line L, we obtain a push-forward measure mu_L possessing many interesting properties, especially when L has "rational" inclination in a certain natural sense. For rational L, the cumulative distribution function F(t) of the projected measure mu_L maps rational numbers into rational numbers, but the converse is not necessarily true. Numerical evidence suggests, in fact, that the inverse image of a rational number is either rational or transcendental. In the special case when L is parallel to one of the heights of S, F(t) is easily described in terms of "weighted" base-2 expansions with weights (1:2) (a natural generalization of standard base-2 expansions which correspond to weights (1:1)). When L is parallel to a side of S, building on work of Bárány, Ferguson and Simon, we compute the local "roughness" of F(t) (i.e., its local Hölder exponents) in terms of logarithms of units in real quadratic fields. Finally, we explain how this interpretation may generalize to other rational lines L.
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