On certain Glasner sets

A sequence of integers S is called Glasner if, given any epsilon > 0 and any infinite subset A of T = R/Z, and given y in T, we can find an integer n is an element of S such that there is an element of {nx : x is an element of A} whose distance to y is not greater than epsilon. In this paper we show that if a sequence of integers is uniformly distributed in the Bohr compactification of the integers, then it is also Glasner. The theorem is proved in a quantitative form.