Univoque numbers and an avatar of Thue-Morse

TL;DR

This paper explores univoque numbers, their unique base expansions, and connects classical results with new findings, including the transcendence of a specific univoque number and the universal appearance of a Thue-Morse avatar.

Contribution

It links historical results on admissible sequences and univoque numbers with new proofs, including transcendence of a particular univoque number and the universal occurrence of a Thue-Morse sequence avatar.

Findings

Identification of the smallest admissible sequence in a set of digits.

Proof that the smallest univoque number in an interval is transcendental.

Demonstration of a Thue-Morse avatar's universal occurrence.

Abstract

Univoque numbers are real numbers $\lambda > 1$ such that the number 1 admits a unique expansion in base $\lambda$, i.e., a unique expansion $1 = \sum_{j \geq 0} a_j \lambda^{-(j+1)}$, with $a_j \in \{0, 1, ..., \lceil \lambda \rceil -1\}$ for every $j \geq 0$. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called {\em admissible sequences}. We show how a 1983 study of the first author gives both a result of Komornik and Loreti on the smallest admissible sequence on the set $\{0, 1, >..., b\}$, and a result of de Vries and Komornik (2007) on the smallest univoque number belonging to the interval $(b, b+1)$, where $b$ is any positive integer. We also prove that this last number is transcendental. An avatar of the Thue-Morse sequence, namely the fixed point beginning in 3 of the morphism $3 \to 31$, $2 \to 30$, $1 \to 03$, $0 \to 02$, occurs in a "universal" manner.