# Approximation properties of β-expansions

Acta Arithmetica (2015)

• Volume: 168, Issue: 3, page 269-287
• ISSN: 0065-1036

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## Abstract

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Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ${\left({ϵ}_{i}\right)}_{i=1}^{\infty }\in {0,1}^{ℕ}$ a β-expansion for x if $x={\sum }_{i=1}^{\infty }{ϵ}_{i}{\beta }^{-i}$. We call a finite sequence ${\left({ϵ}_{i}\right)}_{i=1}^{n}\in {0,1}^{n}$ 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 $\Psi :ℕ\to {ℝ}_{\ge 0}$, we introduce the following subset of ℝ: ${W}_{\beta }\left(\Psi \right):={\bigcap }_{m=1}^{\infty }{\bigcup }_{n=m}^{\infty }{\bigcup }_{{\left({ϵ}_{i}\right)}_{i=1}^{n}\in {0,1}^{n}}\left[{\sum }_{i=1}^{n}\left({ϵ}_{i}\right)/\left({\beta }^{i}\right),{\sum }_{i=1}^{n}\left({ϵ}_{i}\right)/\left({\beta }^{i}\right)+\Psi \left(n\right)\right]$In other words, ${W}_{\beta }\left(\Psi \right)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0\le x-{\sum }_{i=1}^{n}\left({ϵ}_{i}\right)/\left({\beta }^{i}\right)\le \Psi \left(n\right)$. When ${\sum }_{n=1}^{\infty }{2}^{n}\Psi \left(n\right)<\infty$, the Borel-Cantelli lemma tells us that the Lebesgue measure of ${W}_{\beta }\left(\Psi \right)$ is zero. When ${\sum }_{n=1}^{\infty }{2}^{n}\Psi \left(n\right)=\infty$, determining the Lebesgue measure of ${W}_{\beta }\left(\Psi \right)$ is less straightforward. Our main result is that whenever β is a Garsia number and ${\sum }_{n=1}^{\infty }{2}^{n}\Psi \left(n\right)=\infty$ then ${W}_{\beta }\left(\Psi \right)$ is a set of full measure within [0,1/(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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