Monatshefte für Mathematik

, Volume 162, Issue 1, pp 41–60

Growth rate for beta-expansions


DOI: 10.1007/s00605-010-0192-1

Cite this article as:
Feng, DJ. & Sidorov, N. Monatsh Math (2011) 162: 41. doi:10.1007/s00605-010-0192-1


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.


Beta-expansion Bernoulli convolution Pisot number Matrix product Local dimension 

Mathematics Subject Classification (2000)

11A63 28D05 42A85 

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsThe Chinese University of Hong KongShatinPeople’s Republic of China
  2. 2.School of MathematicsThe University of ManchesterManchesterUK