Finite and Infinite Arithmetic Progressions Related to Beta-Expansion

Abstract

Let and $\epsilon (x,\beta )$ be the $\mathrm{\beta }$-expansion of $x\in [\mathrm{0,1})$. Denote by $A_{\beta }(x)$ the set of positions where the digit 1 appears in $\epsilon (x,\beta )$. We consider the sets of points $x$ such that $A_{\beta }(x)$ contains arbitrarily long arithmetic progressions and includes infinite arithmetic progressions, respectively. Their sizes are investigated from the topological, metric, and dimensional viewpoints.