On additive complements. IV

Abstract

Two infinite sequences A and B of non-negative integers are called infinite additive complements, if their sum contains all but finitely many positive integers. Let A(x) and B(x) be the counting functions of A and B, respectively. In 1959, Narkiewicz showed that if A and B are infinite additive complements with A(x)B(x)/x→1 as x→+∞, then A(2x)/A(x)→1 or B(2x)/B(x)→1 as x→+∞. In 1994, Sárközy and Szemerédi confirmed a conjecture of Danzer by proving that, if A and B are infinite additive complements with A(x)B(x)/x→1 as x→+∞, then A(x)B(x)−x→+∞ as x→+∞. In this paper, using Ruzsa's method, we obtain many results on additive complements. For example, we prove that, for infinite additive complements A and B, if limsupA(2x)/A(x)<43, then A(x)B(x)−x→+∞ as x→+∞.