OPEN
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 $1_A\ast 1_A(n) > (\log\frac{4}{3})^{-1}\log n$ (for all large $n$).
