# Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions

### Yann Bugeaud

Université de Strasbourg, France### Bao-Wei Wang

Huazhong University of Science and Technology, Wuhan, China

## Abstract

Let $\beta>1$ be a real number. Let $T_{\beta}$ denote the $\beta$-transformation on $[0,1)$. A cylinder of order $n$ is a set of real numbers in $[0, 1)$ having the same first $n$ digits in their $\beta$-expansion. A cylinder is called full if it has maximal length, i.e., if its length is equal to $\beta^{-n}$. In this paper, we show that full cylinders are well distributed in $[0,1)$ in a suitable sense. As an application to the metrical theory of $\beta$-expansions, we determine the Hausdorff dimension of the set

where $\{z_n\}_{n\ge 1}$ is a sequence of real numbers in $[0,1]$, the function $f:[0,1]\to \R^+$ is continuous, and $S_nf(x)$ denotes the ergodic sum $f(x)+\cdots+f(T^{n-1}_{\beta}x)$.

## Cite this article

Yann Bugeaud, Bao-Wei Wang, Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions. J. Fractal Geom. 1 (2014), no. 2, pp. 221–241

DOI 10.4171/JFG/6