  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 $|A(x)|> \frac 12 (\log x/\log\log x)^2$
for all sufficiently large $x$.
