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 caseαζn≤for small> 0. We are particularly interested in what we call cardinality gap phenomena. For example for fixed ζ > 1 and small> 0 there are at most countably many values of α such thatαζn≤for all large n, whereas largerinduces an uncountable set. We investigate the value ofat 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.