Powers of Rational Numbers Modulo 1 Lying in Short Intervals

Abstract

Let p > q > 1 be two coprime integers. We construct some positive numbers ξ such that the numbers ξ(p/q) n , n = 0, 1, 2, . . . , modulo 1 all lie in a short interval. Our results imply, for instance, that there exist three positive real numbers ξ, ζ, τ such that the inequalities ||ξ(5/3) n || 1/10 and ||τ (3/2)2n || < 14/45 hold for each integer \({n \geqslant 0}\).