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

### Yann Bugeaud

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

Huazhong University of Science and Technology, Wuhan, China

## Abstract

Let $β>1$ be a real number. Let $T_{β}$ denote the $β$-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 $β$-expansion. A cylinder is called full if it has maximal length, i.e., if its length is equal to $β_{−n}$. 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 $β$-expansions, we determine the Hausdorff dimension of the set

where ${z_{n}}_{n≥1}$ is a sequence of real numbers in $[0,1]$, the function $f:[0,1]→R_{+}$ is continuous, and $S_{n}f(x)$ denotes the ergodic sum $f(x)+⋯+f(T_{β}x)$.

## Cite this article

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

DOI 10.4171/JFG/6