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Let    $$\mathcal {H}^{*}=\{h_1,h_2,\ldots \}$$  H ∗ = { h 1 , h 2 , … } be an ordered set of integers. We give sufficient conditions for the existence of increasing sequences of natural numbers    $$a_j$$  a j and    $$n_k$$  n k such that    $$n_k+h_{a_j}$$  n k + h a j is a sum of two squares for every    $$k\ge 1$$  k ≥ 1 and    $$1\le j\le k.$$  1 ≤ j ≤ 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.

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