Counting Beta Expansions and the Absolute Continuity of Bernoulli Convolutions

TL;DR

This paper investigates the growth rate of beta-expansions and establishes new criteria for the absolute continuity of Bernoulli convolutions, linking expansion properties to measure-theoretic regularity.

Contribution

It provides a precise growth rate for beta-expansions when Bernoulli convolutions are absolutely continuous, offering new necessary and sufficient conditions.

Findings

Lower bound for growth rate in general case

Exact growth rate when Bernoulli convolution is absolutely continuous

New criteria for absolute continuity of Bernoulli convolutions

Abstract

We study the typical growth rate of the number of words of length n which can be extended to beta-expansions of x. In the general case we give a lower bound for the growth rate, while in the case that the Bernoulli convolution associated to parameter beta is absolutely continuous we are able to give the growth rate precisely. This gives new necessary and sufficient conditions for the absolute continuity of Bernoulli convolutions.