Unique Representations of Real Numbers in Non-Integer Bases

Glendinning, Paul; Sidorov, Nikita

doi:10.4310/mrl.2001.v8.n4.a12

Cited by 145 publications

(203 citation statements)

References 9 publications

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“…On the other hand, the set of numbers x having a unique expansion has many unexpected topological and combinatorial properties, depending on the value of q; see, e.g., Daróczy and Kátai [1], de Vries [2], [3], Glendinning and Sidorov [7], and [4]. Given a finite alphabet A = {a 1 , .…”

Section: Introductionmentioning

confidence: 99%

Generalized golden ratios of ternary alphabets

Komornik¹,

Lai²,

Pedicini³

2011

J. Eur. Math. Soc.

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Abstract. Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in dependence on the alphabets.

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Section: Introductionmentioning

confidence: 99%

Generalized golden ratios of ternary alphabets

Komornik¹,

Lai²,

Pedicini³

2011

J. Eur. Math. Soc.

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“…Understanding the properties of the set U β is a classical problem within expansions in non-integer bases. In [9] Glendinning and Sidorov showed that U β has strictly positive Hausdorff dimension if β ∈ (β c , 2). Here β c ≈ 1.78723 is the Komornik-Loreti constant introduced in [13].…”

Section: Proof Letmentioning

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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“…For more information on these sets we refer the reader to [7,5,9,6,11,12] and the references therein. Before continuing with our discussion of the sets U α,M and U α,M we make a brief remark.…”

Section: Preliminariesmentioning

confidence: 99%

“…Interesting connections between the size of U α and the Komornik Loreti constant were made in [9]. Using the Thue-Morse sequence we define a new sequence (λ i ) ∈ {−1, 0, 1} N as follows…”

Section: When This Limit Exists Ie D((tmentioning

confidence: 99%

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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Unique Representations of Real Numbers in Non-Integer Bases

Cited by 145 publications

References 9 publications

Generalized golden ratios of ternary alphabets

Generalized golden ratios of ternary alphabets

Approximation properties of -expansions II

Unique expansions and intersections of Cantor sets

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