  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.
