Univoque numbers and an avatar of Thue-Morse

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 $1={\sum }_{j\ge 0}{a}_j{\lambda }^{-(j+1)}$, with $a_j\in \{0,1,...,\lceil \lambda \rceil -1\}$ for every $j\ge 0$. 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.

arXiv commentaccepted by Acta Arithmetica

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