On small bases which admit points with two expansions

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• 作者：Derong Kong^a ; ¹ ; ^{derongkong@126.com} ; Wenxia Li^b ; ^{wxli@math.ecnu.edu.cn} ; Yuru Zou^c ; ^{yuruzou@szu.edu.cn}
• 关键词：11A63 ; 37B10
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：173
• 期：Complete
• 页码：100-128
• 全文大小：489 K
• 卷排序：173

文摘

Given two positive integers M and k  , let Bk(M)Bk(M) be the set of bases q>1q>1 such that there exists a real number x∈[0,M/(q−1)]x∈[0,M/(q−1)] having precisely k different q  -expansions over the alphabet {0,1,…,M}{0,1,…,M}. In this paper we consider k=2k=2 and investigate the smallest base q2(M)q2(M) of B2(M)B2(M). We prove that for M=2mM=2m the smallest base isq2(M)=m+1+m2+2m+52, and for M=2m−1M=2m−1 the smallest base q2(M)q2(M) is the largest positive root ofx4=(m−1)x3+2mx2+mx+1. Moreover, for M=2M=2 we show that q2(2)q2(2) is also the smallest base of Bk(2)Bk(2) for all k≥3k≥3.