On $s$-maximal asymptotic nonbases of density zero

Let $A$ and $G$ be sets of nonnegative integers. The set $A$ is called an asymptotic basis of order $2$ if every sufficiently large integer can be written as the sum of two elements of $A$. However, $A$ is called an asymptotic nonbasis of order 2 if there are infinitely many positive integers that cannot be written as the sum of two elements of $A$. Let $s$ be a positive integer. An asymptotic nonbasis $A$ of order $2$ is $s$-maximal if $A\cup G$ is an asymptotic nonbasis of order $2$ whenever $|G\setminus A| < s$, but $A\cup G$ is an asymptotic basis of order $2$ if $|G\setminus A|\ge s$. We denote by $A(x)$ the number of positive elements of $A$ not exceeding $x$. In 1977, Nathanson constructed an $s$-maximal asymptotic nonbasis $A$ of order 2 such that $A(x)=O(\sqrt{x})$. In this paper, we construct an $s$-maximal asymptotic nonbasis $A$ of order 2 such that $A(x) < 7.887\sqrt{x}$, for all $x\ge 1$.