, Volume 162, Issue 1, pp 41-60
Date: 27 Jan 2010

Growth rate for beta-expansions

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Abstract

Let β > 1 and let m > β be an integer. Each \({x\in I_\beta:=[0,\frac{m-1}{\beta-1}]}\) can be represented in the form $$x=\sum_{k=1}^\infty \epsilon_k\beta^{-k},$$ where \({\epsilon_k\in\{0,1,\ldots,m-1\}}\) for all k (a β-expansion of x). It is known that a.e. \({x\in I_\beta}\) has a continuum of distinct β-expansions. In this paper we prove that if β 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 β. When \({\beta < \frac{1+\sqrt5}2}\) , we show that the set of β-expansions grows exponentially for every internal x.

Communicated by K. Schmidt.