Finite beta-expansions with negative bases

Abstract

The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $\beta$ having the negative finiteness property, that is the set of finite $(-\beta )$-expansions is equal to $\mathbb{Z}[{\beta }^{-1}]$. For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of $(-\beta )$-integers. We also give conditions excluding the negative finiteness property.