2026/07/15 by Ernie Croot, Chi Hoi Yip
#math.NT
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⊆[N], |A|≫√ N, and |Ap|≤ p/2+O(1) for every prime p, then A contains ≫log N elements in the image of a single quadratic. We significantly improve this lower bound to exp(c(√(log N))/(loglog N)). 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.