Erdős Problem #749

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 $n$?

#749: [Er94b]

additive combinatorics

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A similar question can be asked for upper density.

See also [28].

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