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

TL;DR

This paper introduces a new sieve method to find sequences of natural numbers where shifted terms are sums of two squares, with implications for quantum limits on flat tori.

Contribution

It presents a novel modification of the Maynard-Tao sieve and applies it to a prime k-tuples conjecture variant with quantum physics implications.

Findings

Established sufficient conditions for sums of two squares in sequences

Connected number theory results to quantum limits on flat tori

Deduced a conjecture with implications for quantum chaos

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 $k\geq 1$ and $1\leq j\leq k.$ 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.