A power saving for planar unit distances
Source: github.com/openai/math/tree/main/preprints/A-power-saving-f…
We prove a power saving for the planar unit-distance problem: for some absolute $\beta\lt 4/3$, every set of n points in the Euclidean plane determines $O(n^\beta)$ unordered pairs at unit distance.
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According to the Oct 6 Open AI document dump, the upper bound can be lowered from $O(n^{4/3})$ to $O(n^{4/3-\delta})$ for some positive (not explicitly computed) $\delta$ . Source here ; I have not checked its accuracy.
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreGil KalaiOct 7, 2026
includes the sharp-order bound for distinct distances in every fixed dimension and an improved planar unit-distance bound for some absolute . Erdős’s arithmetic-progression conjecture. My post on Bloom and Sisask’s breakthrough discussed the three-term case of the assertion that divergent…