A variation of the prime k-tuples conjecture with applications to quantum limits

Abstract

Let mathcal {H}^{*}={h_1,h_2,ldots } be an ordered set of integers. We give sufficient conditions for the existence of increasing sequences of natural numbers a_j and n_k such that n_k+h_{a_j} is a sum of two squares for every kge 1 and 1le jle k. Our method uses a novel modification of the Maynard–Tao sieve together with a second moment estimate. As a special case of our result, we deduce a conjecture due to D. Jakobson which has several implications for quantum limits on flat tori.