Problem Statement
Let $A\subset\mathbb{N}$ be the set of cubes. Is it true that\[1_A\ast 1_A(n) \ll (\log n)^{O(1)}?\]
Categories:
Number Theory
Progress
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 $n$. Stewart [St08] improved this to\[1_A\ast 1_A(n) \gg (\log n)^{11/13}.\]Source: erdosproblems.com/829 | Last verified: January 16, 2026