# Approximation properties of β-expansions

Acta Arithmetica (2015)

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

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topSimon 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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