OPEN
This is open, and cannot be resolved with a finite computation.
Let $f(N)$ be maximal such that there exists $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=\lfloor N^{1/2}\rfloor$ such that $\lvert (A+A)\cap [1,N]\rvert=f(N)$. Estimate $f(N)$.
Erdős and Freud
[ErFr91] proved\[\left(\frac{3}{8}-o(1)\right)N \leq f(N) \leq \left(\frac{1}{2}+o(1)\right)N,\]and note that it is closely connected to the size of the largest quasi-Sidon set (see
[840]).
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Additional thanks to: Terence Tao
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #819, https://www.erdosproblems.com/819, accessed 2026-01-16
