Erdős Problem #871

Let \(A\) be an additive basis of order \(2\), and suppose \(1_A\ast 1_A(n)\to \infty\) as \(n\to \infty\). Can \(A\) be partitioned into two disjoint additive bases of order \(2\)? A question of Erdős and Nathanson [ErNa88], who proved this is true if \(1_A\ast 1_A(n) > c\log n\) (for all large \(n\)) for some constant \(c>(\log\frac{4}{3})^{-1}\). Erdős and Nathanson [ErNa89] also proved that for every \(t\) there exists a basis \(A\) of order \(2\) such that \(1_A\ast 1_A(n)\geq t\) for all large \(n\) and yet \(A\) cannot be partitioned into two disjoint additive bases. This has been disproved by Larsen using Claude Opus 4.5 - in fact only a small modification of the argument of [ErNa89] is required.