Almost Every Number Has a Continuum of <i>β</i>-Expansions

Sidorov, Nikita

doi:10.1080/00029890.2003.11920025

Cited by 93 publications

(112 citation statements)

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“…One aspect of these representations that makes them interesting is that for α ∈ ( 1 M +1 , 1) a generic x ∈ I α,M has many α-expansions (cf. [4,21,22]). This naturally leads researchers to study the set of x ∈ I α,M with a unique α-expansion, the so called univoque set.…”

Section: Preliminariesmentioning

confidence: 99%

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Unique expansions and intersections of Cantor sets

Baker

Kong

2017

Nonlinearity

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Abstract. To each α ∈ (1/3, 1/2) we associate the Cantor setIn this paper we consider the intersection Γ α ∩ (Γ α + t) for any translation t ∈ R. We pay special attention to those t with a unique {−1, 0, 1} α-expansion, and study the setWe prove that there exists a transcendental number α KL ≈ 0.39433 . . . such that: D α is finite for α ∈ (α KL , 1/2), D αKL is infinitely countable, and D α contains an interval for α ∈ (1/3, α KL ). We also prove that D α equals [0, log 2 − log α ] if and only if α ∈ (1/3, 3− √ 5 2 ]. As a consequence of our investigation we prove some results on the possible values of dim H (Γ α ∩ (Γ α + t)) when Γ α ∩ (Γ α + t) is a self-similar set. We also give examples of t with a continuum of {−1, 0, 1} α-expansions for which we can explicitly calculate dim H (Γ α ∩ (Γ α + t)), and for which Γ α ∩ (Γ α + t) is a self-similar set. We also construct α and t for which Γ α ∩ (Γ α + t) contains only transcendental numbers.Our approach makes use of digit frequency arguments and a lexicographic characterisation of those t with a unique {−1, 0, 1} α-expansion.

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Section: Preliminariesmentioning

confidence: 99%

“…In fact it can be shown that Lebesgue almost every t ∈ Γ α − Γ α has a continuum of α-expansions (cf. [4,21,22]). Thus within the parameter space (1/3, 1/2) we are forced to have the following more complicated interpretation of Γ α ∩ (Γ α + t) (cf.…”

Section: Introductionmentioning

confidence: 99%

Unique expansions and intersections of Cantor sets

Baker

Kong

2017

Nonlinearity

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“…For typical x the β-expansion of x is not unique, indeed almost every x ∈ [ −1 β−1 , 1 β−1 ] has uncountably many β-expansions, see [8]. This allows one, given x, to search for β-expansions of x with interesting properties, such as a given digit frequency or that the sequence is a β-expansion of x for more than one β.…”

Section: Introductionmentioning

confidence: 99%

Self-affine sets with positive Lebesgue measure

Dajani

Jiang

Kempton

2014

Indagationes Mathematicae

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“…Despite being a simple generalisation of the well known integer base expansions, β-expansions exhibit very different behaviour. As an example, a well known theorem of Sidorov [19] states that for any β ∈ (1, 2), almost every x ∈ I β has a continuum of β-expansions. This is of course completely different to the usual integer base expansions, where every number has a unique expansion except for a countable set of exceptions which have precisely two.…”

Section: Introductionmentioning

confidence: 99%

Approximation properties of -expansions II

Baker¹

2017

Ergod. Th. Dynam. Sys.

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Let $\unicode[STIX]{x1D6FD}\in (1,2)$ be a real number. For a function $\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{\geq 0}$, define $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ to be the set of $x\in \mathbb{R}$ such that for infinitely many $n\in \mathbb{N},$ there exists a sequence $(\unicode[STIX]{x1D716}_{i})_{i=1}^{n}\in \{0,1\}^{n}$ satisfying $0\leq x-\sum _{i=1}^{n}(\unicode[STIX]{x1D716}_{i}/\unicode[STIX]{x1D6FD}^{i})\leq \unicode[STIX]{x1D6F9}(n)$. In Baker [Approximation properties of $\unicode[STIX]{x1D6FD}$-expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$, the condition $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)=\infty$ implies that $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers $(\unicode[STIX]{x1D714}_{n})_{n=1}^{\infty }$ satisfying $\lim _{n\rightarrow \infty }\unicode[STIX]{x1D714}_{n}=\infty$, for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1.497,\ldots ,2)$, the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D714}_{n}\cdot 2^{-n})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. We also study the case where $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)<\infty$ in which the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ has Lebesgue measure zero. Applying the mass transference principle developed by Beresnevich and Velani in [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], we obtain some results on the Hausdorff dimension and the Hausdorff measure of $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$.

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Almost Every Number Has a Continuum of β-Expansions

Cited by 93 publications

References 6 publications

Unique expansions and intersections of Cantor sets

Unique expansions and intersections of Cantor sets

Self-affine sets with positive Lebesgue measure

Approximation properties of -expansions II

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