Suppose $\mathscr{A} \subset \mathbb{N}$ enjoys for all large enough cutoffs $X$ the following properties:

- $|\mathscr{A} \cap [1,X]| > \sqrt{X}/10$; and
- the discriminant $\prod_{\alpha \neq \beta}|\alpha-\beta|$ over $\alpha,\beta \in \mathscr{A} \cap [1,X]$ is composed only of primes smaller than $10\sqrt{X}$.

Is there a quadratic $an^2 + bn+c$, with $a > 0$, such that $\mathscr{A} \subset \{an^2+bn +c \mid n \in \mathbb{N} \}$?

*Notes*. The values of any such quadratic has both properties, upon replacing the number $10$ with an appropriate constant. The question is whether those are essentially all examples. It is motivated by a similar unsolved problem in the so-called "inverse large sieve," in which condition two is replaced by a sifting condition $|\mathscr{A} \mod{p}| \leq (p+1)/2$ for all primes $p$ (and the same conclusion is expected to hold up to finitely many exceptions); cf. Ben Green and Adam Harper's recent GAFA paper, *Inverse questions for the large sieve*. I am interested even in a proof that $\liminf |\mathscr{A} \cap [1,X]|/\sqrt{X} < \infty$.