Additive completion of thin sets (Q6564328)

We say that positive integer sets \(A\) and \(B\) are exact additive complements if \(A+B\) contains all sufficiently large integers and \(A(x)B(x)/x\to\infty\). Let \(A(x)\) denote the counting function of \(A\) and let \(a^*(x)\) denote the largest element in \(A\cap[1, x]\). The authors proved that the following nice result, for exact additive complements \(A\) and \(B\) with \(a_{n+1}/na_{n}\to\infty\), \[A(x)B(x)-x\geq\frac{a^*(x)}{A(x)}+o\left(\frac{a^*(x)}{A^2(x)}\right) ~~\text{as}~~x\to\infty.\] Moreover, they also construct exact additive complements \(A\) and \(B\) with \(a_{n+1}/na_{n}\to\infty\) such that \[A(x)B(x)-x\leq\frac{a^*(x)}{A(x)}+(1+o(1))\left(\frac{a^*(x)}{A^2(x)}\right) \] for infinitely many positive integers \(x\).