A property of Pisot numbers and Fourier transforms of self-similar measures

Abstract

For any Pisot number β it is known that the set F(β) = {t: lim _n→∞‖tβ ⁿ‖ = 0} is countable, where ‖a‖ is the distance between a real number a and the set of integers. In this paper it is proved that every member in this set is of the form cβ ⁻ⁿ, where n is a nonnegative integer and c is determined by a linear system of equations. Furthermore, for some self-similar measures µ associated with β, the limit at infinity of the Fourier transforms $\lim _{n \to \infty } \hat \mu (t\beta ^n ) \ne 0$ if and only if t is in a certain subset of F(β). This generalizes a similar result of Huang and Strichartz.