Exceptional digit frequencies and expansions in non-integer bases

TL;DR

This paper investigates how digit frequency distributions in non-integer base expansions evolve as the base approaches 1, revealing they fill out the entire simplex, with implications for Bernoulli convolutions.

Contribution

It establishes that digit frequencies in $eta$-expansions fill the simplex as $eta$ approaches 1, providing new insights into digit distribution and local dimensions.

Findings

Digit frequencies fill the simplex as $eta$ approaches 1

Upper bounds for local dimensions of biased Bernoulli convolutions

Characterization of digit frequency sets in non-integer bases

Abstract

In this paper we study the set of digit frequencies that are realised by elements of the set of $\beta$-expansions. The main result of this paper demonstrates that as $\beta$ approaches $1,$ the set of digit frequencies that occur amongst the set of $\beta$-expansions fills out the simplex. As an application of our main result, we obtain upper bounds for the local dimension of certain biased Bernoulli convolutions.