On the distance from a rational power to the nearest integer

We prove that for any non-zero real number ξ the sequence of fractional parts {ξ(3/2)ⁿ}, n=1,2,3,…, contains at least one limit point in the interval 0.238117…,0.761882…] of length 0.523764…. More generally, it is shown that every sequence of distances to the nearest integer ||ξ(p/q)ⁿ||, n=1,2,3,…, where p/q>1 is a rational number, has both ‘large’ and ‘small’ limit points. All obtained constants are explicitly expressed in terms of p and q. They are also expressible in terms of the Thue-Morse sequence and, for irrational ξ, are best possible for every pair p>1, q=1. Furthermore, we strengthen a classical result of Pisot and Vijayaraghavan by giving similar effective results for any sequence ||ξαⁿ||, n=1,2,3,…, where α>1 is an algebraic number and where ξ≠0 is an arbitrary real number satisfying ξ∉Q(α) in case α is a Pisot or a Salem number.