On small univoque bases of real numbers

TL;DR

This paper characterizes the smallest base for which a positive real number has a unique expansion in binary-like form, especially focusing on bases less than or equal to the Komornik-Loreti constant, and identifies exact conditions for equality.

Contribution

It provides a complete characterization of numbers with minimal univoque bases below the Komornik-Loreti constant and determines explicit values of the base in those cases.

Findings

Identifies when the minimal univoque base equals the Komornik-Loreti constant.

Provides explicit formulas for the minimal univoque base when it is less than the Komornik-Loreti constant.

Characterizes the set of numbers for which the minimal univoque base reaches the critical constant.

Abstract

Given a positive real number $x$, we consider the smallest base $q_s(x)\in(1,2)$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that \[ x=\sum_{i=1}^\infty\frac{d_i}{(q_s(x))^i}. \] In this paper we give complete characterizations of those $x$'s for which $q_s(x)\le q_{KL}$, where $q_{KL}$ is the Komornik-Loreti constant. Furthermore, we show that $q_s(x)=q_{KL}$ if and only if \[ x\in\left\{1, ~\frac{q_{KL}}{q_{KL}^2-1},~ \frac{1}{q_{KL}^2-1}, ~\frac{1}{q_{KL}(q_{KL}^2-1)}\right\}. \] Finally, we determine the explicit value of $q_s(x)$ if $q_s(x)<q_{KL}$.