Problem Statement
For what functions $g(N)\to \infty$ is it true that\[\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}\]implies $\limsup 1_A\ast 1_A(n)=\infty$?
Categories:
Number Theory Additive Basis
Progress
This is a stronger form of the Erdős-Turán conjecture [28] (since establishing this for any function $g(N)\to \infty$ would imply a positive solution to [28]).Source: erdosproblems.com/40 | Last verified: January 13, 2026