Smallest bases of expansions with multiple digits

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 $\{0,1,\cdots ,M\}$. In this paper we investigate the smallest base $q_2$ of $\B_2$, and show that if $M=2m$ the smallest base $$q_2=\frac{m+1+\sqrt{m^2+2m+5}}{2},$$ and if $M=2m-1$ the smallest base $q_2$ is the appropriate root of $$x^4=(m-1)x^3+2m{x}^2+mx+1.$$ Moreover, for $M=2$ we show that $q_2$ is also the smallest base of $\B_k$ for all $k\ge 3$. This turns out to be different from that for $M=1$.

27 pages

References in corpus (5)

• Expansions in non-integer bases: lower, middle and top orders
• Greedy and quasi-greedy expansions in non-integer bases
• On univoque Pisot numbers
• On a problem of countable expansions
• Numbers with countable expansions in base of generalized golden ratios