Additive completition of thin sets

Abstract: Two sets $A,B$ of positive integers are called \emph{exact additive complements}, if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow1$. Let $A={a_1<a_2<\cdots}$ be a set of positive integers. Denote $A(x)$ by the counting function of $A$ and $a^*(x)$ by the largest element in $A\bigcap [1,x]$. Following the work of Ruzsa and Chen-Fang, we prove that, for exact additive complements $A,B$ with $\frac{a_{n+1}}{na_n}\rightarrow\infty$, we have $A(x)B(x)-x\ge \frac{a^(x)}{A(x)}+o\left(\frac{a^(x)}{A(x)^2}\right)$ as $x\rightarrow +\infty$. On the other hand, we also construct exact additive complements $A,B$ with $\frac{a_{n+1}}{na_n}\rightarrow\infty$ such that $A(x)B(x)-x\le \frac{a^(x)}{A(x)}+(1+o(1))\left(\frac{a^(x)}{A(x)^2}\right)$ holds for infinitely many positive integers $x$.