Inverse questions for the large sieve

Abstract

Suppose that an infinite set $${\fancyscript{A}}$$ 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 $${\fancyscript{A}}$$ 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 $${\fancyscript{A}}$$ have some additive structure. Unfortunately we cannot solve the problem itself.