Integral Powers of Numbers in Small Intervals Modulo 1: The Cardinality Gap Phenomenon

This paper deals with the distribution of αζⁿ mod 1, where α ≠ 0, ζ > 1 are fixed real numbers and n runs through the positive integers. Denote by ‖·‖ the distance to the nearest integer. We investigate the case of αζⁿ all lying in prescribed small intervals modulo 1 for all large n, with focus on the case ‖αζⁿ‖ ≤ ɛ for small ɛ > 0. We are particularly interested in what we call cardinality gap phenomena. For example for fixed ζ > 1 and small ɛ > 0 there are at most countably many values of α such that ‖αζⁿ‖ ≤ ɛ for all large n, whereas larger ɛ induces an uncountable set. We investigate the value of ‖ at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational ζ > 1. Results concerning Pisot and Salem numbers such as some contribution to Mahler’s 3/2-problem are implicitly deduced. We study similar questions for fixed α ≠ 0 as well.