# 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.

## How to cite

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Simon Baker. "Approximation properties of β-expansions." Acta Arithmetica 168.3 (2015): 269-287. <http://eudml.org/doc/279768>.

@article{SimonBaker2015,
abstract = {Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence $(ϵ_\{i\})_\{i=1\}^\{∞\} ∈ \{0,1\}^\{ℕ\}$ a β-expansion for x if $x=∑_\{i=1\}^\{∞\}ϵ_\{i\}β^\{-i\}$. We call a finite sequence $(ϵ_\{i\})_\{i=1\}^\{n\} ∈ \{0,1\}^\{n\}$ 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 $Ψ: ℕ → ℝ_\{≥ 0\}$, we introduce the following subset of ℝ: $W_\{β\}(Ψ) := ⋂ _\{m=1\}^\{∞\} ⋃ _\{n=m\}^\{∞\} ⋃ _\{(ϵ_\{i\})_\{i=1\}^\{n\}∈\{0,1\}^\{n\}\} [∑_\{i=1\}^\{n\} (ϵ_\{i\})/(β^\{i\}),∑_\{i=1\}^\{n\}(ϵ_\{i\})/(β^\{i\}) + Ψ(n)]$In other words, $W_\{β\}(Ψ)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0 ≤ x - ∑_\{i=1\}^\{n\} (ϵ_\{i\})/(β^\{i\}) ≤ Ψ(n)$. When $∑_\{n=1\}^\{∞\} 2^\{n\}Ψ(n) < ∞$, the Borel-Cantelli lemma tells us that the Lebesgue measure of $W_\{β\}(Ψ)$ is zero. When $∑_\{n=1\}^\{∞\}2^\{n\}Ψ(n) = ∞$, determining the Lebesgue measure of $W_\{β\}(Ψ)$ is less straightforward. Our main result is that whenever β is a Garsia number and $∑_\{n=1\}^\{∞\}2^\{n\}Ψ(n) = ∞$ then $W_\{β\}(Ψ)$ 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.},
author = {Simon Baker},
journal = {Acta Arithmetica},
keywords = {beta-expansion; garsia number; Bernoulli convolution},
language = {eng},
number = {3},
pages = {269-287},
title = {Approximation properties of β-expansions},
url = {http://eudml.org/doc/279768},
volume = {168},
year = {2015},
}

TY - JOUR
AU - Simon Baker
TI - Approximation properties of β-expansions
JO - Acta Arithmetica
PY - 2015
VL - 168
IS - 3
SP - 269
EP - 287
AB - Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence $(ϵ_{i})_{i=1}^{∞} ∈ {0,1}^{ℕ}$ a β-expansion for x if $x=∑_{i=1}^{∞}ϵ_{i}β^{-i}$. We call a finite sequence $(ϵ_{i})_{i=1}^{n} ∈ {0,1}^{n}$ 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 $Ψ: ℕ → ℝ_{≥ 0}$, we introduce the following subset of ℝ: $W_{β}(Ψ) := ⋂ _{m=1}^{∞} ⋃ _{n=m}^{∞} ⋃ _{(ϵ_{i})_{i=1}^{n}∈{0,1}^{n}} [∑_{i=1}^{n} (ϵ_{i})/(β^{i}),∑_{i=1}^{n}(ϵ_{i})/(β^{i}) + Ψ(n)]$In other words, $W_{β}(Ψ)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0 ≤ x - ∑_{i=1}^{n} (ϵ_{i})/(β^{i}) ≤ Ψ(n)$. When $∑_{n=1}^{∞} 2^{n}Ψ(n) < ∞$, the Borel-Cantelli lemma tells us that the Lebesgue measure of $W_{β}(Ψ)$ is zero. When $∑_{n=1}^{∞}2^{n}Ψ(n) = ∞$, determining the Lebesgue measure of $W_{β}(Ψ)$ is less straightforward. Our main result is that whenever β is a Garsia number and $∑_{n=1}^{∞}2^{n}Ψ(n) = ∞$ then $W_{β}(Ψ)$ 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.
LA - eng
KW - beta-expansion; garsia number; Bernoulli convolution
UR - http://eudml.org/doc/279768
ER -

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