Diophantine approximations with Pisot numbers

Abstract

Let α be a Pisot number. Let L(α) be the largest positive number such that for some ξ = ξ(α) ∈ R the limit points of the sequence of fractional parts {ξα n } ∞ n=1 all lie in the interval [L(α), 1 − L(α)]. In this paper we show that if α is of degree at most 4 or α ≤ √ 5+1 2then L(α) ≥ 3 17 . Also we find explicitly the value of L(α) for certain Pisot numbers of degree 3.