Approximation properties of $\beta$-expansions

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 $\Psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$, we introduce the following subset of $\mathbb{R}$, $$W_{\beta}(\Psi):=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^\infty\bigcup_{(\epsilon_{i})_{i=1}^{n}\in\{0,1\}^{n}}\Big[\sum_{i=1}^{n}\frac{\epsilon_i}{\beta^{i}}, \sum_{i=1}^n\frac{\epsilon_i} {\beta^i}+\Psi(n)\Big]$$ In other words, $W_{\beta}(\Psi)$ is the set of $x\in\mathbb{R}$ for which there exists infinitely many solutions to the inequalities $$0\leq x-\sum_{i=1}^{n}\frac{\epsilon_{i}}{\beta^{i}}\leq \Psi(n).$$ 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.