OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but o(n) points of any n-point planar set determine at least n^{1-ε} distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by O(n^{4/3-δ}) for an absolute δ > 0.
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We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but o(n) points of any n-point planar set determine at least n^{1-ε} distinct nonzero distances.
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We prove a power saving for the planar unit-distance problem: for some absolute β< 4/3, every set of n points in the Euclidean plane determines O(n^β) unordered pairs at unit distance.
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