A sharp inverse theorem for the quadratic large sieve

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 bo