On the fractional parts of powers of Pisot numbers of length at most 4

Abstract

The aim of this paper is twofold. We first give a list of all Pisot polynomials of length at most 4. It contains seven polynomials of degree at most 5, and two infinite series of polynomials with unbounded degree. Then, for Pisot numbers α of length 3 and 4, we find explicitly the largest positive number L(α) such that for some ξ=ξ(α)∈R the limit points of the sequence of fractional parts {ξαn}n=1∞ all lie in the interval [L(α),1−L(α)].