ON THE FRACTIONAL PARTS OF RATIONAL POWERS

Abstract

Let ξ be a non-zero real number, and let a = p/q > 1 be a rational number. We denote by U(a,ξ) and L(a,ξ) the largest and the smallest limit points of the sequence of fractional parts {ξ an}, n = 0,1,2,…, respectively. A possible way to prove Mahler's conjecture claiming that Z-numbers do not exist is to show that U(3/2,ξ) > 1/2 for every ξ > 0. We prove that U(3/2,ξ) cannot belong to [0,1/3) ∪ S, where S is an explicit infinite union of intervals in (1/3,1/2). This result is a corollary to a more general result claiming that, for any rational a > 1, U(a,ξ) cannot lie in a certain union of intervals. We also obtain new inequalities for the difference U(a,ξ) - L(a,ξ). Using them we show that some analogues of Z-numbers do not exist.