Restricted Research - Award List, Note/Discussion Page

Fiscal Year: 2021

175  University of North Texas  (84471)

Principal Investigator: Allaart,Pieter

Total Amount of Contract, Award, or Gift (Annual before 2011): $ 6,000

Exceeds $250,000 (Is it flagged?): No

Start and End Dates: - 8/31/25

Restricted Research: YES

Academic Discipline: Mathematics

Department, Center, School, or Institute: College of Science

Title of Contract, Award, or Gift: Non-integer base expansions and multifractal analysis

Name of Granting or Contracting Agency/Entity: Simons Foundation


Program Title: N/A


In 2006, H. Okamoto introduced a one-parameter family {F_a} of self-affine functions which includes the Cantor function and examples of Perkins and Katsuura. Okamoto proved that, depending on the value of the parameter a, his function is either (i) nowhere differentiable; (ii) nondifferentiable almost everywhere but differentiable at uncountably many points; or (iii) differentiable almost everywhere but nondifferentiable at uncountably many points. In a 2016 paper teh PI computed the Hausdorff dimension of the exceptional sets in cases (ii) and (iii), and characterized the set of points where F_a has an infinite derivative. This last set is closely related to the set A_b of numbers having a unique expansion in base b (the "univoque set"), where b=1/a. This connection inspired me to write a paper on beta-expansions (published in 2017) in which I introduced a subset of the univoque set called "strongly univoque set". THe PI determined for which values of the base b the strongly univoque set is equal to the univoque set, and showed that when it is not, the difference W_b between the two sets is uncountable, but its Hausdorff dimension is sometimes zero, sometimes strictly positive. Finally, in a 2018 paper the PI extended my results on Okamoto's function by computing the Holder (or multifractal) spectrum of a large class of self-affine functions. SInce then, the PI has extended these results further to include self-affine functions involving vertical shears. Jaerisch and Sumi, inspired by my paper, extended the result in a different direction.

Discussion: No discussion notes


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