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Approximation properties of β-expansions

100%

Baker S.

nr 3

269-287

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.

On univoque points for self-similar sets

51%

Baker S. , Dajani K. , Jiang K.

nr 3

265-282

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.