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

TL;DR

This paper investigates the distribution modulo 1 of sequences formed by polynomial evaluations at powers of Salem numbers, providing computational methods and explicit density functions for degree 4 Salem numbers.

Contribution

It introduces a computational approach to analyze the distribution of polynomial sequences at Salem numbers and derives explicit density functions for degree 4 cases.

Findings

Sequence $( heta^n)$ modulo 1 is dense but not uniformly distributed.

Explicit density functions are obtained for degree 4 Salem numbers.

Computational methods confirm theoretical results.

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.