On additive complements. II

Abstract:

Two infinite sequences $A$ 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 $B$ and let $\limsup \limits _{x\rightarrow \infty }A(x)B(x)/ x$ $=\alpha (A, B)$. Recently, the authors [Proceedings of the American Mathematical Society 138 (2010), 1923-1927] proved that for additive complements $A$ and $B$, if $\alpha (A, B)<5/4$ or $\alpha (A, B)>2$, then $A(x)B(x)-x\rightarrow +\infty$ as $x\to \infty$. In this paper, we prove that for any $\varepsilon >0$ there exist additive complements $A$ and $B$ with $2-\varepsilon <\alpha (A, B) <2$ and $A(x)B(x)-x=1$ for infinitely many positive integers $x$.