Problem 749

Erdős Problem #749: declared status 'open'. Formalized: yes. Let $\epsilon>0$. Does there exist $A\subseteq \mathbb{N}$ such that the lower density of $A+A$ is at least $1-\epsilon$ and yet $1_A\ast 1_A(n) \ll_\epsilon 1$ for all $n$? Prize: no. Tags: additive combinatorics.