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

Abstract Let $$\mathcal {H}^{*}=\{h_1,h_2,\ldots \}$$ H ∗ = { h 1 , 2 … } be an ordered set of integers. We give sufficient conditions for the existence increasing sequences natural numbers $$a_j$$ a j and $$n_k$$ n k such that $$n_k+h_{a_j}$$ + is a sum two squares every $$k\ge 1$$ ≥ $$1\le j\le k.$$ ≤ . Our method uses novel modification Maynard–Tao sieve together with second moment estimate. As special case our result, we deduce conjecture due to D. Jakobson which has several implications quantum limits on flat tori.