On the smallest base in which a number has a unique expansion

TL;DR

This paper investigates the minimal base in which a positive real number has a unique expansion, providing explicit formulas, algorithms, and analyzing the structure and properties of the associated level sets and the function defining the minimal base.

Contribution

It offers a comprehensive characterization of the minimal base for unique expansions, including explicit descriptions, an efficient algorithm, and a detailed analysis of the function's continuity and the structure of level sets.

Findings

The minimal base function is right-continuous with no downward jumps.

Almost all points have a finite level set, but some have infinitely many.

The level set at the Komornik-Loreti constant has infinitely many accumulation points.

Abstract

Given a real number $x>0$, we determine $q_s(x):=\inf\mathscr{U}(x)$, where $\mathscr{U}(x)$ is the set of all bases $q\in(1,2]$ for which $x$ has a unique expansion of $0$'s and $1$'s. We give an explicit description of $q_s(x)$ for several regions of $x$-values. For others, we present an efficient algorithm to determine $q_s(x)$ and the lexicographically smallest unique expansion of $x$. We show that the infimum is attained for almost all $x$, but there is also a set of points of positive Hausdorff dimension for which the infimum is proper. In addition, we show that the function $q_s$ is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of $q_s$. A large part of the paper is devoted to the level sets $L(q):=\{x>0:q_s(x)=q\}$. We show that $L(q)$ is finite for almost every $q$, but there are also infinitely many infinite level sets. In particular, for the Komornik-Loreti constant $q_{KL}=\min\mathscr{U}(1)\approx 1.787$ we prove that $L(q_{KL})$ has both infinitely many left- and infinitely many right accumulation points.