On small bases which admit points with two expansions

Derong Kong; Wenxia Li; Yuru Zou

doi:10.1016/j.jnt.2016.09.012

On small bases which admit points with two expansions

Derong Kong
, Wenxia Li
, Yuru Zou^*

^*Corresponding author for this work

School of Mathematical Sciences

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

10 Scopus citations

Abstract

Given two positive integers M and k, let B_k(M) be the set of bases q<1 such that there exists a real number x∈[0,M/(q−1)] having precisely k different q-expansions over the alphabet {0,1,…,M}. In this paper we consider k=2 and investigate the smallest base q₂(M) of B₂(M). We prove that for M=2m the smallest base is q₂(M)=+1+2+2+52, and for M=2m−1 the smallest base q₂(M) is the largest positive root of x⁴=(m−1)x³+2mx²+mx+1. Moreover, for M=2 we show that q₂(2) is also the smallest base of B_k(2) for all k≥3.

Original language	English
Pages (from-to)	100-128
Number of pages	29
Journal	Journal of Number Theory
Volume	173
DOIs	https://doi.org/10.1016/j.jnt.2016.09.012
State	Published - 1 Apr 2017

Keywords

Beta expansions
Smallest bases
Two expansions
Unique expansion

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10.1016/j.jnt.2016.09.012

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title = "On small bases which admit points with two expansions",

abstract = "Given two positive integers M and k, let Bk(M) be the set of bases q<1 such that there exists a real number x∈[0,M/(q−1)] having precisely k different q-expansions over the alphabet \{0,1,…,M\}. In this paper we consider k=2 and investigate the smallest base q2(M) of B2(M). We prove that for M=2m the smallest base is q2(M)=+1+2+2+52, and for M=2m−1 the smallest base q2(M) is the largest positive root of x4=(m−1)x3+2mx2+mx+1. Moreover, for M=2 we show that q2(2) is also the smallest base of Bk(2) for all k≥3.",

keywords = "Beta expansions, Smallest bases, Two expansions, Unique expansion",

author = "Derong Kong and Wenxia Li and Yuru Zou",

note = "Publisher Copyright: {\textcopyright} 2016 Elsevier Inc.",

year = "2017",

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day = "1",

doi = "10.1016/j.jnt.2016.09.012",

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TY - JOUR

T1 - On small bases which admit points with two expansions

AU - Kong, Derong

AU - Li, Wenxia

AU - Zou, Yuru

PY - 2017/4/1

Y1 - 2017/4/1

N2 - Given two positive integers M and k, let Bk(M) be the set of bases q<1 such that there exists a real number x∈[0,M/(q−1)] having precisely k different q-expansions over the alphabet {0,1,…,M}. In this paper we consider k=2 and investigate the smallest base q2(M) of B2(M). We prove that for M=2m the smallest base is q2(M)=+1+2+2+52, and for M=2m−1 the smallest base q2(M) is the largest positive root of x4=(m−1)x3+2mx2+mx+1. Moreover, for M=2 we show that q2(2) is also the smallest base of Bk(2) for all k≥3.

AB - Given two positive integers M and k, let Bk(M) be the set of bases q<1 such that there exists a real number x∈[0,M/(q−1)] having precisely k different q-expansions over the alphabet {0,1,…,M}. In this paper we consider k=2 and investigate the smallest base q2(M) of B2(M). We prove that for M=2m the smallest base is q2(M)=+1+2+2+52, and for M=2m−1 the smallest base q2(M) is the largest positive root of x4=(m−1)x3+2mx2+mx+1. Moreover, for M=2 we show that q2(2) is also the smallest base of Bk(2) for all k≥3.

KW - Beta expansions

KW - Smallest bases

KW - Two expansions

KW - Unique expansion

UR - https://www.scopus.com/pages/publications/84995561809

U2 - 10.1016/j.jnt.2016.09.012

DO - 10.1016/j.jnt.2016.09.012

M3 - 文章

AN - SCOPUS:84995561809

SN - 0022-314X

VL - 173

SP - 100

EP - 128

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

On small bases which admit points with two expansions

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