Freiman's theorem answers a question of Erdős (Q1092105)

Let A be a set of positive integers and \(A_ 2=\{a_ 1+a_ 2:\) \(a_ i\in A\}\). Let A(x) and \(A_ 2(x)\) be the counting functions of A and \(A_ 2\), respectively. A question of Erdős is whether or not \(\lim_{x\to \infty}A_ 2(x)/A(x)=\infty\) when A is a basis and \(A(x)=o(x)\). The author shows that a theorem of \textit{G. A. Freiman} [see p. 54 of ``Foundations of a structural theory of set addition'', Transl. Math. Monogr. 37 (Providence 1973; Zbl 0271.10044) (Russian original Kazan' 1966; Zbl 0203.353)] implies an affirmative answer to Erdős' question.