On uniformly distributed sequences of integers and Poincaré recurrence

Abstract

We give a general result true for locally compact groups which in the special case of the integers says that if a sequence of natural numbers ( k n ) n = 1 ∞ is uniformly distributed on Z and for each irrational number α the sequence (< k n α >) n = 1 ∞ is uniformly distributed modulo one, then ( k n ) n = 1 ∞ is a set of Poincaré recurrence. We then show there are indeed a number families of sequences with this property.