Approximation properties of $\beta$-expansions

Abstract

Let $\beta \in (1,2)$ and $x\in [0,\frac{1}{\beta -1}]$. We call a sequence $({\epsilon }_i{)}_{i=1}^{\infty }\in \{0,1{\}}^{\mathbb{N}}$ a $\beta$-expansion for $x$ if $x={\sum }_{i=1}^{\infty }{\epsilon }_i{\beta }^{-i}$. We call a finite sequence $({\epsilon }_i{)}_{i=1}^n\in \{0,1{\}}^n$ an $n$-prefix for $x$ if it can be extended to form a $\beta$-expansion of $x$. In this paper we study how good an approximation is provided by the set of $n$-prefixes. Given $\Psi :\mathbb{N}\to {\mathbb{R}}_{\ge 0}$, we introduce the following subset of $\mathbb{R}$, $$W_{\beta }(\Psi ):=\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }\bigcup _{({\epsilon }_i{)}_{i=1}^n\in \{0,1{\}}^n}[\sum _{i=1}^n\frac{{\epsilon }_i}{{\beta }^i},\sum _{i=1}^n\frac{{\epsilon }_i}{{\beta }^i}+\Psi (n)]$$ In other words, $W_{\beta }(\Psi )$ is the set of $x\in \mathbb{R}$ for which there exists infinitely many solutions to the inequalities $$0\le x-\sum _{i=1}^n\frac{{\epsilon }_i}{{\beta }^i}\le \Psi (n).$$ When ${\sum }_{n=1}^{\infty }{2}^n\Psi (n)<\infty$ the Borel-Cantelli lemma tells us that the Lebesgue measure of $W_{\beta }(\Psi )$ is zero. When ${\sum }_{n=1}^{\infty }{2}^n\Psi (n)=\infty ,$ determining the Lebesgue measure of $W_{\beta }(\Psi )$ is less straightforward. Our main result is that whenever $\beta$ is a Garsia number and ${\sum }_{n=1}^{\infty }{2}^n\Psi (n)=\infty$ then $W_{\beta }(\Psi )$ is a set of full measure within $[0,\frac{1}{\beta -1}]$. Our approach makes no assumptions on the monotonicity of $\Psi ,$ unlike in classical Diophantine approximation where it is often necessary to assume $\Psi$ is decreasing.