Approximation properties of $\beta $-expansions

Simon Baker¹

¹ School of Mathematics The University of Manchester Oxford Road Manchester M13 9PL, United Kingdom

Acta Arithmetica, Tome 168 (2015) no. 3, pp. 269-287.

Résumé

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.

Détail

DOI : 10.4064/aa168-3-4

Keywords: beta beta call sequence epsilon infty mathbb beta expansion sum infty epsilon beta i call finite sequence epsilon n prefix extended form beta expansion paper study approximation provided set n prefixes given psi mathbb mathbb geq introduce following subset mathbb beta psi bigcap infty bigcup infty bigcup epsilon biggl sum frac epsilon beta sum frac epsilon beta psi biggr other words beta psi set mathbb which there exist infinitely many solutions inequalities leq x sum frac epsilon beta leq psi sum infty psi infty borel cantelli lemma tells lebesgue measure beta psi zero sum infty psi infty determining lebesgue measure beta psi straightforward main result whenever beta garsia number sum infty psi infty beta psi set full measure within beta approach makes assumptions monotonicity psi unlike classical diophantine approximation where often necessary assume psi decreasing