Beta-representations of 0 and Pisot numbers.
Christiane Frougny, Edita Pelantová · arXiv (Cornell University) · 2015
Let $\beta >1 $, $d$ a positive integer, and $$Z_{\beta,d}=\{z_{1} z_{2}\cdots \mid \sum_{i\ge 1}z_i \beta^{-i}=0, \; z_i \in \{-d, \ldots, d\}\}$$ be the set of infinite words having value 0 in base $\beta$ on the alphabet $\{-d, \ldots, d\}$. Based on a recent result of Feng on spectra of numbers, we prove that if the set $Z_{\beta,\lceil \beta \rceil -1}$ is recognizable by a finite B\uchi automaton then $\beta$ is a Pisot number. As a consequence of previous results, the set $Z_{\beta, d}$ is recognizable by a finite B\uchi automaton for every positive integer $d$ if and only if $Z_{\beta, d}$ is recognizable by a finite B\uchi automaton for one $d \ge \lceil \beta \rceil -1$. These conditions are equivalent to the fact that $\beta$ is a Pisot number. The bound $\lceil \beta \rceil -1$ cannot be further reduced.