Bases which admit exactly two expansions

Yi Cai; Wenxia Li

doi:10.5486/PMD.2023.9471

Bases which admit exactly two expansions

Yi Cai
, Wenxia Li

School of Mathematical Sciences

Shanghai Institute of Technology

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

Abstract

Given a positive integer m, let Ωm = {0, 1, . . ., m}, and let B2(m) denote the set of bases q ∈ (1, m + 1] in which there exist numbers having precisely two q-expansions over the alphabet Ωm. Sidorov [23] firstly studied the set B2(1) and raised some questions. Komornik and Kong [15] further investigated the set B2(1) and partially answered Sidorov’s questions. In the present paper, we consider the set B2(m) for general positive integer m, and generalise the results obtained by Komornik and Kong.

Original language	English
Pages (from-to)	339-384
Number of pages	46
Journal	Publicationes Mathematicae Debrecen
Volume	103
Issue number	3-4
DOIs	https://doi.org/10.5486/PMD.2023.9471
State	Published - 2023

Keywords

generalized golden ratio
q-expansion
quasi-greedy q-expansion
unique q-expansion

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title = "Bases which admit exactly two expansions",

abstract = "Given a positive integer m, let Ωm = \{0, 1, . . ., m\}, and let B2(m) denote the set of bases q ∈ (1, m + 1] in which there exist numbers having precisely two q-expansions over the alphabet Ωm. Sidorov [23] firstly studied the set B2(1) and raised some questions. Komornik and Kong [15] further investigated the set B2(1) and partially answered Sidorov{\textquoteright}s questions. In the present paper, we consider the set B2(m) for general positive integer m, and generalise the results obtained by Komornik and Kong.",

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AU - Cai, Yi

AU - Li, Wenxia

PY - 2023

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N2 - Given a positive integer m, let Ωm = {0, 1, . . ., m}, and let B2(m) denote the set of bases q ∈ (1, m + 1] in which there exist numbers having precisely two q-expansions over the alphabet Ωm. Sidorov [23] firstly studied the set B2(1) and raised some questions. Komornik and Kong [15] further investigated the set B2(1) and partially answered Sidorov’s questions. In the present paper, we consider the set B2(m) for general positive integer m, and generalise the results obtained by Komornik and Kong.

AB - Given a positive integer m, let Ωm = {0, 1, . . ., m}, and let B2(m) denote the set of bases q ∈ (1, m + 1] in which there exist numbers having precisely two q-expansions over the alphabet Ωm. Sidorov [23] firstly studied the set B2(1) and raised some questions. Komornik and Kong [15] further investigated the set B2(1) and partially answered Sidorov’s questions. In the present paper, we consider the set B2(m) for general positive integer m, and generalise the results obtained by Komornik and Kong.

KW - generalized golden ratio

KW - q-expansion

KW - quasi-greedy q-expansion

KW - unique q-expansion

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SP - 339

EP - 384

JO - Publicationes Mathematicae Debrecen

JF - Publicationes Mathematicae Debrecen

IS - 3-4

ER -

Bases which admit exactly two expansions

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Keywords

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