PROVED
This has been solved in the affirmative.
For any $M\geq 1$, if $A\subset \mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\in A+A$ such that $a+1,a-1\not\in A+A$.
There may even be $\gg \lvert A\rvert^2$ many such $a$. A similar question can be asked for truncations of infinite Sidon sets.
This is true; in fact there are at least $(1-o(1))\lvert A+A\rvert$ many such $a\in A+A$. A proof of (a slightly weaker statement) was found by DeepMind using an elementary argument. Independently, Bloom noted (see the comments) that in fact this strong statement already follows from an argument in
[ESS94], which shows that, for any $d\neq 0$,\[\lvert (A+A)\cap (A+A+d)\rvert \leq 2\lvert A\rvert^{3/2}.\]
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This page was last edited 03 April 2026. View history
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Additional thanks to: Cedric Pilatte and GTsoukalas
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 #152, https://www.erdosproblems.com/152, accessed 2026-04-11
