Beta-representations of 0 and Pisot numbers

Let β > 1, d a positive integer, and Zβ,d = {z1z2 · · · | ∑ i>1 ziβ −i = 0, zi ∈ {−d, . . . , d}} be the set of infinite words having value 0 in base β on the alphabet {−d, . . . , d}. Based on a recent result of Feng on spectra of numbers, we prove that if the set Zβ,⌈β⌉−1 is recognizable by a finite Büchi automaton then β is a Pisot number. As a consequence of previous results, the set Zβ,d is recognizable by a finite Büchi automaton for every positive integer d if and only if Zβ,d is recognizable by a finite Büchi automaton for one d > ⌈β⌉− 1. These conditions are equivalent to the fact that β is a Pisot number. The bound ⌈β⌉ − 1 cannot be further reduced.