The quadratic inverse sieve conjecture

Let $N$ be a positive integer and let $A\subseteq [N]$. For each prime $p$, define the set of occupied residue classes

$$A_p=\{a\pmod p:a\in A\}.$$

A rational quadratic is the image of a quadratic polynomial with rational coefficients. The quadratic inverse sieve conjecture. If

$$|A_p|\leq (p+1)/2$$

for every prime $p$ and $|A|\gg\sqrt N$, then a positive proportion of $A$ should lie in the image of a rational quadratic.

The conjecture asserts that squares and, more generally, integer values of rational quadratics account for all near-equality cases in the quadratic large-sieve bound. It is an open inverse problem in sieve theory.