OPEN - $500

Is there $A\subseteq \mathbb{N}$ such that
\[\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}\]
exists and is $\neq 0$?

A suitably constructed random set has this property if we are allowed to ignore an exceptional set of density zero. The challenge is obtaining this with no exceptional set. Erdős believed the answer should be no. Erdős and Sárkzözy proved that
\[\frac{\lvert 1_A\ast 1_A(n)-\log n\rvert}{\sqrt{\log n}}\to 0\]
is impossible. Erdős suggests it may even be true that the $\liminf$ and $\limsup$ of $1_A\ast 1_A(n)/\log n$ are always separated by some absolute constant.