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 $A\subseteq [N]$, $\mathrm{\mid }A\mathrm{\mid }\gg \sqrt{N}$, and $\mathrm{\mid }{A}_p\mathrm{\mid }\le p\mathrm{/}2+O(1)$ 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 $$\exp \left(c\frac{\sqrt{\log N}}{\log \log N}\right)\mathrm{.}$$ 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.