Univoque sets for real numbers

• Autorzy: Fan Lü , Bo Tan , Jun Wu
• Języki publikacji: EN

Abstrakty

EN

For x ∈ (0,1), the univoque set for x, denoted 𝒰(x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with $x_{i} ∈ {0,1}$. We prove that for any x ∈ (0,1), 𝒰(x) contains a sequence ${β_{k}}_{k ≥ 1}$ increasing to 2. Moreover, 𝒰(x) is a Lebesgue null set of Hausdorff dimension 1; both 𝒰(x) and its closure $\overline {𝒰(x)}$ are nowhere dense.

Kategorie tematyczne

• 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2014
• Tom: 227
• Numer: 1
• Strony: 69-83

Daty

wydano

2014

Twórcy

autor

Fan Lü

• School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074, Wuhan, P.R. China

autor

Bo Tan

• School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074, Wuhan, P.R. China

autor

Jun Wu

• School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074, Wuhan, P.R. China

Identyfikatory

DOI

10.4064/fm227-1-5