On the distribution modulo 1 of the sum of powers of a Salem number

Abstract

It is well known that the sequence of powers of a Salem number θ, modulo 1, is dense in the unit interval, but is not uniformly distributed. Generalizing a result of Dupain, we determine, explicitly, the repartition function of the sequence (P(θn)mod 1)n≥1, where P is a polynomial with integer coefficients and θ is quartic. Also, we consider some examples to illustrate the method of determination.