A sharp inverse theorem for the quadratic large sieve

Abstract

We prove that if $A\subseteq [N]$, $|A|\gg \sqrt{N}$, and $|A_p|\le p/2+O(1)$ for every prime $p\ll \log N$, then $A$ contains at least $\exp \left(c\sqrt{\log N}/\log \log N\right)$ elements in the image of a single integral quadratic $m\pm x^2$. This significantly improves Hanson's logarithmic lower bound, while requiring the residue restriction only for primes $p\ll \log N$. Our proof uses a new weighted entropy argument inspired by our previous work \href{https://arxiv.org/abs/2606.17487}{arXiv:2606.17487} with Sheffer. A matching construction shows that this exponential scale is optimal even when the quadratic may be chosen arbitrarily in $\mathbb{Z}[x]$. We also show extending the residue restrictions to primes up to $(\log N{)}^3$ yields the stronger bound $\exp (c\sqrt{\log N/\log \log N})$ via a Selberg-sieve-type argument, and discuss an application to the inverse Goldbach problem.

22 pages, title updated, authors added. This version contains stronger results