OPEN
Let $A\subset\mathbb{N}$ be the set of cubes. Is it true that
\[1_A\ast 1_A(n) \ll (\log n)^{O(1)}?\]
Mordell proved that
\[\limsup_{n\to \infty} 1_A\ast 1_A(n)=\infty\]
and Mahler
[Ma35b] proved
\[1_A\ast 1_A(n) \gg (\log n)^{1/4}\]
for infinitely many $n$.
