Approximation properties of $\beta $-expansions
Tom 168 / 2015
Streszczenie
Let $\beta\in(1,2)$ and $x\in [0,1/(\beta-1)]$. 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 $\mathbb{R}$: $$W_{\beta}(\Psi):=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^{\infty}\bigcup_{(\epsilon_{i})_{i=1}^{n}\in\{0,1\}^{n}}\biggl[\sum_{i=1}^{n}\frac{\epsilon_{i}}{\beta^{i}},\sum_{i=1}^{n}\frac{\epsilon_{i}}{\beta^{i}}+\Psi(n)\biggr]$$ In other words, $W_{\beta}(\Psi)$ is the set of $x\in\mathbb{R}$ for which there exist 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 $[0,1/(\beta-1)]$. 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.
