Let f(n) be a function from the integers to the set of nonnegative integers together with infinity, and suppose that f(n) has only finitely many zeros. Sidon sets will be used to prove that there exists a set A of integers such that A has representation function f(n) and A contains at least x^c elements a with |a| > x and c > 0. This solves a well-known problem in additive number theory.

Mel Nathanson