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1

Approximation properties of β-expansions

100%

Baker S.

Acta Arithmetica

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2015

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tom 168

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nr 3

269-287

EN

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.

2

The growth rate and dimension theory of beta-expansions

100%

Baker S.

Fundamenta Mathematicae

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2012

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tom 219

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nr 3

271-285

EN

In a recent paper of Feng and Sidorov they show that for β ∈ (1,(1+√5)/2) the set of β-expansions grows exponentially for every x ∈ (0,1/(β-1)). In this paper we study this growth rate further. We also consider the set of β-expansions from a dimension theory perspective.

3

On univoque points for self-similar sets

51%

Baker S., Dajani K., Jiang K.

Fundamenta Mathematicae

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2015

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tom 228

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nr 3

265-282

EN

Let K ⊆ ℝ be the unique attractor of an iterated function system. We consider the case where K is an interval and study those elements of K with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams (1988). The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of this set. Our algorithm can be applied generically, and our result generalises the work of Daróczy, Kátai, Kallós, Komornik and de Vries.

Ograniczanie wyników

2 Fundamenta Mathematicae

1 Acta Arithmetica

3 Baker S.

1 Dajani K.

1 Jiang K.

2 2015

1 2012

Approximation properties of β-expansions

The growth rate and dimension theory of beta-expansions

On univoque points for self-similar sets