Inverse questions for the large sieve

Suppose that an infinite set A occupies at most 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.