A weighted entropy approach for the quadratic inverse large sieve conjecture

The quadratic inverse large sieve problem predicts that the examples sharp at the square-root threshold are essentially quadratic. Hanson proved the first unconditional result in this direction: if $A\subseteq[N]$, $|A|\gg\sqrt N$, and $|A_p|\le p/2+O(1)$ for every prime $p$, then $A$ contains $\gg\log N$ elements in the image of a single quadratic. We significantly improve this lower bound to \[
\exp\left(c\frac{\sqrt{\log N}}{\log\log N}\right). \] We also prove density-dependent variants, including a two-set version motivated by Green--Harper's robust inverse large sieve conjectures and their connection with the inverse Goldbach problem. Combined with a theorem of Elsholtz--Harper on hypothetical decompositions of the primes, our results show that any such decomposition would force both summands to have large intersections with quadratic images. Our proof combines a weighted entropy argument with sieve estimates, inspired by the recent work of Croot--Mao--Pohoata--Sheffer--Yip.