The Erdős distinct distances problem in $\mathbb{R}^3$

Jonathan Tidor, Hung-Hsun Hans Yu, Dmitrii Zakharov

arXiv:2608.14454v1 Primary: math.CO Secondary: math.MG

Abstract

We prove that $N$ points in $\mathbb{R}^3$ determine at least $N^{2/3-o(1)}$ distinct distances.

Disclosure

“nstitute for the Theory of Computing. We thank SLMath, Matija Bucić, AIM, and the Simons Institute for their hospitality. AI use statement. The mathematical content of this paper and all of the writing were generated solely by the authors. ChatGPT-5.6 was used during the editorial process to identify mathematical and typographical errors. 2. Proof overview 2.1. Choosing coordinates. In Section 3, we choose coordinates on E + and E − , the s”
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Verified: 2026-08-23T05:15:38.008952Z

Structural counts

Pages 147 pdf

Theorems 30 source

Lemmas 40 source

Propositions 28 source

Corollaries 9 source

Definitions 38 source

Displayed equations 497 source

Bibliography entries 100 source

Appendix pages 146 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.