Approximation properties of β-expansions

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 -