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

Abstract

Let $\theta $ be a Salem number and $P(x)$ a polynomial with integer coefficients. It is well-known that the sequence $(\theta^n)$ modulo 1 is dense but not uniformly distributed. In this article we discuss the sequence $(P(\theta^n))$ modulo 1. Our first approach is computational and consists in estimating the number of n so that the fractional part of $(P(\theta^n))$ falls into a subinterval of the partition of $[0,1]$. If Salem number is of degree 4 we can obtain explicit density function of the sequence, using an algorithm which is also given. Some examples confirm that these two approaches give the same result.