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? ? (? ) = supc? [0, 1):? x? (Qp? Z)? [0, c)? , d (x) is purely periodic.
Ben et al. (Wed,) studied this question.