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Erdős problem 749

Let ϵ>0\epsilon>0. Does there exist ANA\subseteq \mathbb{N} such that the lower density of A+AA+A is at least 1ϵ1-\epsilon and yet 1A1A(n)ϵ11_A\ast 1_A(n) \ll_\epsilon 1 for all nn?

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