Remarks on the disproof of the unit distance conjecture

Noga Alon; Thomas F. Bloom; W. T. Gowers; Daniel Litt; Will Sawin; Arul Shankar; Jacob Tsimerman; Victor Wang; Melanie Matchett Wood

Remarks on the disproof of the unit distance conjecture

Combinatorics 2026-05-21 v1 Number Theory

Authors: Noga Alon , Thomas F. Bloom , W. T. Gowers , Daniel Litt , Will Sawin , Arul Shankar , Jacob Tsimerman , Victor Wang , Melanie Matchett Wood

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Abstract

We present a short, digested, human-verified version of the recent OpenAI-generated counterexample to the Erd\H{o}s unit distance conjecture, and a sequence of reflections on it. The argument relies crucially on ideas that may, at least in retrospect, be attributed to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.

Keywords

unit distance conjecture Erdős problem counterexample additive combinatorics polynomial methods

Cite

@article{arxiv.2605.20695,
  title  = {Remarks on the disproof of the unit distance conjecture},
  author = {Noga Alon and Thomas F. Bloom and W. T. Gowers and Daniel Litt and Will Sawin and Arul Shankar and Jacob Tsimerman and Victor Wang and Melanie Matchett Wood},
  journal= {arXiv preprint arXiv:2605.20695},
  year   = {2026}
}

Comments

19 pages; expository note based on very recent developments

Remarks on the disproof of the unit distance conjecture

Abstract

Keywords

Cite

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