On infinite additive complements

Abstract

Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. For a sequence T of non-negative integers, let T(x) be the number of terms of T not exceeding x. In 1994, Sarkozy and Szemeredi confirmed a conjecture of Danzer by proving that, for infinite additive complements A and B, if lim sup A(x)B(x)/x ≤ 1, then A(x)B(x)−x → +∞ as x → +∞. In this paper, it is proved that, if A and B are infinite additive complements with lim sup A(x)B(x)/x < (4√2 + 2)/7 = 1.093 · · ·, then (A(x)B(x) − x)/min{A(x);B(x)} → +∞ as x → +∞.