GLASNER SETS AND POLYNOMIALS IN PRIMES

A set of integers S is said to be Glasner if for every infinite subset A of the torus T = R/Z and epsilon > 0 there exists some n is an elem ent of S such that the dilation nA = {nx: x is an element of A} inters ects every integral of length epsilon in T. In this paper we show that if p(n) denotes the nth prime integer and f is any non-constant polyn omial mapping the natural numbers to themselves, then (f(p(n)))(n grea ter than or equal to 1) is Glasner. The theorem is proved in a quantit ative form and generalizes a result of Alon and Peres (1992).