How Small Must Ill-distributed Sets Be?

Consider a set S ⊂ Z. Suppose that, for many primes p, the distribution of S in congruence classes mod p is far from uniform. How sparse is S forced to be thereby? A clear dichotomy appears: it seems that S must either be very small or possess much algebraic structure. We show that, if S ⊂ Z ∩ [0, N ] occupies few congruence classes mod p for many p, then either S has fewer than N elements or most of S is contained in an algebraic curve of degree O (1). Similar statements are conjectured for S ⊂ Z, n 6= 2. We follow an approach that combines ideas from the larger sieve of Gallagher [Ga] and from the work of Bombieri and Pila [BP]. All techniques used are elementary.