Problem 749

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Statement. 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 $n$?

Status. Open.

Source. erdosproblems.com/749, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #749, https://www.erdosproblems.com/749.

References.

• [Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.

Formalization. Statement in formal-conjectures.

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• erdos_1994_some_problems_number_theory_combinatorics_combinatorial_geometry