Inverse questions for the large sieve

Abstract: Suppose that an infinite set $A$ occupies at most $\frac{1}{2}(p+1)$ residue classes modulo $p$, for every sufficiently large prime $p$. The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of $A$ that are at most $X$ is $O(X^{1/2})$, and the quadratic examples show that this is sharp. The simplest form of the inverse large sieve problem asks whether they are the only examples. We prove a variety of results and formulate various conjectures in connection with this problem, including several improvements of the large sieve bound when the residue classes occupied by $A$ have some additive structure. Unfortunately we cannot solve the problem itself.