Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 183-01. A power saving for planar halving lines
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OpenAI@2026-10-07#rem:oai-p-183-01
Authors: OpenAI. Family 183.
Title: A power saving for planar halving lines.
Abstract: There are absolute constants ε > 0 and C such that every sufficiently large even n-point set in the plane with no three collinear has at most $Cn{}{4/3-{}varepsilon}$ unordered halving pairs. This gives a power saving over the classical $O(n{}{4/3})$ bound for planar halving lines. The proof is nonquantitative and does not supply explicit constants.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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