Approximation properties of $β$-expansions

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

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