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

For any Pisot number ?? it is known that the set F(??) = {t: lim _n??????t?? 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 c?? ^?n , where n is a nonnegative integer and c is determined by a linear system of equations. Furthermore, for some self-similar measures ?? associated with ??, the limit at infinity of the Fourier transforms $lim _{n to infty } hat mu (tbeta ^n ) ne 0$ if and only if t is in a certain subset of F(??). This generalizes a similar result of Huang and Strichartz.