A sequence of integers S is called Glasner if, given any ε > 0 and any infinite subset A of T = R/Z, and given y in T, we can find an integer n ∈ S such that there is an element of {nx : x ∈ A} whose distance to y is not greater than ε. 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.