On finite additive complements

Two sets A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let A(x) and B(x) be the counting functions of A and B. Up to now, all researches on additive complements are under the assumption that A and B are infinite sets. In this paper, we consider the case that A is a finite set. This is very different from the infinite case.