β-expansions of rational numbers with Pisot Chabauty basis in Qp

The aim of this paper is to study some arithmetic properties about the periodicity of the ?-expansion of p-adic numbers. We prove that for every Pisot Chabauty unit number such that the finiteness property (F) is satisfied, there exists a constant ??(?) for which every rational in [0, ??(?)[ have a purely periodic ?-expansion, where ??(?) = sup{c ? [0, 1): ?x ? (Qp ? Z) ? [0, c)?, d (x) is purely periodic}.