Smallest bases of expansions with multiple digits

TL;DR

This paper determines the smallest bases for which real numbers have exactly two or more expansions over a digit set, revealing explicit formulas depending on the digit set size and showing special cases for different parameters.

Contribution

It provides explicit formulas for the smallest bases with exactly two expansions for various digit set sizes, extending understanding of base expansions with multiple digits.

Findings

For even M=2m, the smallest base q_2 is (m+1+√(m^2+2m+5))/2.

For odd M=2m-1, q_2 is the root of a specific quartic polynomial.

For M=2, q_2 is also the smallest base for all k≥3.

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 $$ x^4=(m-1)\,x^3+2 m\, x^2+m \,x+1. $$ 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 $M=1$.