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

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 $f:[0,1]\to {\mathbb{R}}^{+}$ is continuous, and $S_{n}f(x)$ denotes the ergodic sum $f(x)+\cdots +f(T_{\beta }^{n-1}x)$.