Diophantine approximations with Pisot numbers

الأصل · EN

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 {ξαⁿ}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 + 1/2 then L(α)≥ 3/17. Also we find explicitly the value of L(α) for certain Pisot numbers of degree 3.