On a conjecture of additive complements

Abstract

Abstract For two infinite sequences A and B of non-negative integers, if every sufficiently large integer can be expressed as the sum of two elements taken from A and B, then we call such A, B additive complements. Let A(x) (resp. B(x)) be the counting functions of A (resp. B). Motivated by a result of Sárközy and Szemerédi, the authors proved that if limsupA(x)B(x)/x<3−3 or limsupA(x)B(x)/x>2, then A(x)B(x)−x→+∞ as x→+∞. Afterwards we also posed a natural conjecture that: for additive complements A, B, there exists a set T of non-negative integers with density one such that A(x)B(x)−x→+∞asx∈Tandx→+∞. In this paper, we confirm this conjecture. Furthermore, we obtain a stronger evaluation for the special additive complements.