Growth rate for beta-expansions

TL;DR

This paper investigates the growth rate of beta-expansions for Pisot numbers, revealing a universal growth rate for almost every point and linking it to local dimensions of Bernoulli convolutions, with exponential growth for certain beta values.

Contribution

It establishes a universal growth rate for beta-expansions when beta is a Pisot number and connects this rate to the local dimension of Bernoulli convolutions, providing new insights into their structure.

Findings

For Pisot beta, a.e. x has a unique growth rate of beta-expansions.

The growth rate is linked to the Lebesgue-generic local dimension of Bernoulli convolutions.

When beta< (1+√5)/2, the number of beta-expansions grows exponentially for all x.

Abstract

Let $\beta>1$ and let $m>\be$ be an integer. Each $x\in I_\be:=[0,\frac{m-1}{\beta-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty \epsilon_k\beta^{-k}, \] where $\epsilon_k\in\{0,1,...,m-1\}$ for all $k$ (a $\beta$-expansion of $x$). It is known that a.e. $x\in I_\beta$ has a continuum of distinct $\beta$-expansions. In this paper we prove that if $\beta$ is a Pisot number, then for a.e. $x$ this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by $\beta$. When $\beta<\frac{1+\sqrt5}2$, we show that the set of $\beta$-expansions grows exponentially for every internal $x$.