Algebraic sums and products of univoque bases

Karma Dajani; Vilmos Komornik; Derong Kong; Wenxia Li

doi:10.1016/j.indag.2018.05.010

Algebraic sums and products of univoque bases

Karma Dajani
, Vilmos Komornik
, Derong Kong^*
, Wenxia Li

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

Given x∈(0,1], let U(x) be the set of bases q∈(1,2] for which there exists a unique sequence (d_i) of zeros and ones such that x=∑_i=1 ^∞d_i∕qⁱ. Lü et al. (2014) proved that U(x) is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum U(x)+λU(x) and product U(x)⋅U(x)^λ contain an interval for all x∈(0,1] and λ≠0. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (Dajani and Kalle, 2017).

Original language	English
Pages (from-to)	1087-1104
Number of pages	18
Journal	Indagationes Mathematicae
Volume	29
Issue number	4
DOIs	https://doi.org/10.1016/j.indag.2018.05.010
State	Published - Aug 2018

Keywords

Algebraic differences
Cantor sets
Non-integer base expansions
Non-matching parameters
Thickness
Univoque bases

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10.1016/j.indag.2018.05.010

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@article{317349ffd8ac42a5b768be680c9e39c5,

title = "Algebraic sums and products of univoque bases",

abstract = "Given x∈(0,1], let U(x) be the set of bases q∈(1,2] for which there exists a unique sequence (di) of zeros and ones such that x=∑i=1 ∞di∕qi. L{\"u} et al. (2014) proved that U(x) is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum U(x)+λU(x) and product U(x)⋅U(x)λ contain an interval for all x∈(0,1] and λ≠0. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (Dajani and Kalle, 2017).",

keywords = "Algebraic differences, Cantor sets, Non-integer base expansions, Non-matching parameters, Thickness, Univoque bases",

author = "Karma Dajani and Vilmos Komornik and Derong Kong and Wenxia Li",

note = "Publisher Copyright: {\textcopyright} 2018 Royal Dutch Mathematical Society (KWG)",

year = "2018",

month = aug,

doi = "10.1016/j.indag.2018.05.010",

language = "英语",

volume = "29",

pages = "1087--1104",

journal = "Indagationes Mathematicae",

issn = "0019-3577",

publisher = "Elsevier B.V.",

number = "4",

}

TY - JOUR

T1 - Algebraic sums and products of univoque bases

AU - Dajani, Karma

AU - Komornik, Vilmos

AU - Kong, Derong

AU - Li, Wenxia

PY - 2018/8

Y1 - 2018/8

N2 - Given x∈(0,1], let U(x) be the set of bases q∈(1,2] for which there exists a unique sequence (di) of zeros and ones such that x=∑i=1 ∞di∕qi. Lü et al. (2014) proved that U(x) is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum U(x)+λU(x) and product U(x)⋅U(x)λ contain an interval for all x∈(0,1] and λ≠0. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (Dajani and Kalle, 2017).

AB - Given x∈(0,1], let U(x) be the set of bases q∈(1,2] for which there exists a unique sequence (di) of zeros and ones such that x=∑i=1 ∞di∕qi. Lü et al. (2014) proved that U(x) is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum U(x)+λU(x) and product U(x)⋅U(x)λ contain an interval for all x∈(0,1] and λ≠0. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (Dajani and Kalle, 2017).

KW - Algebraic differences

KW - Cantor sets

KW - Non-integer base expansions

KW - Non-matching parameters

KW - Thickness

KW - Univoque bases

UR - https://www.scopus.com/pages/publications/85047978059

U2 - 10.1016/j.indag.2018.05.010

DO - 10.1016/j.indag.2018.05.010

M3 - 文章

AN - SCOPUS:85047978059

SN - 0019-3577

VL - 29

SP - 1087

EP - 1104

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

IS - 4

ER -

Algebraic sums and products of univoque bases

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Keywords

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