Problem 868

Erdős Problem #868: declared status 'solved'. Formalized: yes. If $A$ is an additive basis of order $2$, and $1_A\ast 1_A(n)\to \infty$ as $n\to \infty$, then must $A$ contain a minimal additive basis of order $2$? (i.e. such that deleting any element creates infinitely many $n\not\in A+A$) What if $1_A\ast 1_A(n) >\epsilon \log n$ (for all large $n$, for arbitrary fixed $\epsilon>0$)? Current best: Erd\H{o}s and Nathanson [ErNa89] proved that, for any $t$, there exists $A$ such that $1_A\ast 1_A(n)\geq t$ for all large $n$ and yet $A$ does not contain a minimal asymptotic basis of order $2$. Prize: no. Tags: additive basis, number theory.