Treffer: Independence of solution sets in additive number theory

Let \(A\subseteq {\mathbb{N}}\), then \(A\) is called an asymptotic basis of order 2 if for all sufficiently large \(n\in {\mathbb{N}}\) there are \(a,a'\in A\) such that \(n=a+a'\). Let \(S_A(n)=\{a\in A|\) \(n-a\in A\), \(n\neq 2a\}\) denote the solution set of \(n\). By the Erdős-Rényi probabilistic method [see \textit{H.Halberstam} and \textit{K.F.Roth}, Sequences (1966; Zbl 0141.04405), p. 141 ff.] it is shown that for almost all \(A\) in the space \(\Omega\) of all strictly increasing sequences of positive integers the cardinality of \(S_A(m)\cap S_A(n)\) is bounded for all \(m