Let \(A\) be an infinite set of positive integers with the property that at most finitely many integers have exactly one representation in the form \(a+a'\), \(a\leq a'\), \(a,a'\in A\). \textit{J.-L. Nicolas}, \textit{I. Z. Ruzsa}, and \textit{A. Sárközy} [J. Number Theory 73, 292--317 (1998; Zbl 0921.11050)] proved that such sets must satisfy \(A(x)> c ( \log x / \log \log x)^{3/2}\) infinitely often and constructed such a set satisfying \(A(x)\ll ( \log x)^2\). In this paper the lower estimate is improved to \(A(x)\gg ( \log x / \log \log x)^2\), so now the lower and upper estimates differ only by an iterated logarithm.