Planar distinct distances and unit-distance bounds

What OpenAI claims

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. (Lean)

Papers (2)

1. The weak pinned planar distance theorem

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

2. A power saving for planar unit distances

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.