ADDITIVE COMPLETION OF THIN SETS

Two sets $A,B$ of positive integers are called exact additive complements if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow 1$ . For $A=\{a_1<a_2<\cdots \}$ , let $A(x)$ denote the counting function of A and let $a^*(x)$ denote the largest element in $A\bigcap [1,x]$ . Following the work of Ruzsa [‘Exact additive complements’, Quart. J. Math.68 (2017) 227–235] and Chen and Fang [‘Additive complements with Narkiewicz’s condition’, Combinatorica39 (2019), 813–823], we prove that, for exact additive complements $A,B$ with ${a_{n+1}}/ {na_n}\rightarrow \infty $ , $$ \begin{align*}A(x)B(x)-x\geqslant \frac{a^*(x)}{A(x)}+o\bigg(\frac{a^*(x)}{A(x)^2}\bigg) \quad\mbox{as } x\rightarrow +\infty.\end{align*} $$ We also construct exact additive complements $A,B$ with ${a_{n+1}}/{na_n}\rightarrow \infty $ such that $$ \begin{align*}A(x)B(x)-x\leqslant \frac{a^*(x)}{A(x)}+(1+o(1))\bigg(\frac{a^*(x)}{A(x)^2}\bigg)\end{align*} $$ for infinitely many positive integers x.