On additive representation functions

Let $A$ be an infinite set of natural numbers. For $n\in\mathbb{N}$, let $r(A,n)$ denote the number of solutions of the equation $n=a+b$ with $a,b\in A$, $a\le b$. Let $|A(x)|$ be the number of integers in $A$ which are less than or equal to $x$. In this paper, we prove that if $r(A,n)\not=1$ for all sufficiently large integers $n$, then $|A(x)|>\frac12(\log x/\log\log x)^2$ for all sufficiently large $x$.