Smallest bases of expansions with multiple digits

2015/07/29 by Kong, Derong, Li, Wenxia, Zou, Yuru
#11A63 #37B10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1507.08135

Abstract

Given two positive integers M and k, let \B_k be the set of bases q>1 such that there exists a real number x having exactly k different q-expansions over the alphabet \0,1,⋯,M\. In this paper we investigate the smallest base q₂ of \B₂, and show that if M=2m the smallest base q₂ =(m+1+√(m²+2m+5))/(2), and if M=2m-1 the smallest base q₂ is the appropriate root of x⁴=(m-1) x³+2 m x²+m x+1. Moreover, for M=2 we show that q₂ is also the smallest base of \B_k for all k≥ 3. This turns out to be different from that for M=1.

Related

• Multiple expansions of real numbers with digits set \0,1,q\
• On small bases which admit countably many expansions
• Bases which admit exactly two expansions
• Expansions in multiple bases
• Optimal expansions in non-integer bases