On the distance from a rational power to the nearest integer

Abstract

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 | | ξ α n | | , 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.