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 $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and $\varepsilon>0$ there exists some $n\in S$ such that the dilation $nA=\{nx\colon x\in A\}$ intersects every integral of length $\varepsilon$ in $\mathbb{T}$. In this paper we show that if $p_n$ denotes the $n$th prime integer and $f$ is any non-constant polynomial mapping the natural numbers to themselves, then $(f(p_n))_{n\geq 1}$ is Glasner. The theorem is proved in a quantitative form and generalizes a result of Alon and Peres (1992).