If $A$ is an additive basis 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 $\epsilon>0$)?
