The Minkowski grid has robustly many repeated distances
Lee, Sungchul, Pohoata, Cosmin, Zhu, Daniel G.arXiv.orgPublished Jul 6, 2026Original︎ ↗Archived copy
Stewards’ notes · 1
- The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
Supports. An independent construction by different authors, inspired by the number-field counterexample but proceeding through a robust Ramanujan-type estimate; the counterexample follows by taking the whole set. Only the abstract was read on this pass; the proof itself was not checked here. Sep 16, 2026
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Text57 passages · 1 marked
Taking $A=P$, the inequality above gives a distance occurring $n^{1+\delta}$ times in $P$; thereby a scaled copy of $P$ is a counterexample for the unit-distance conjecture. In addition, the same inequality shows that (1) all subsets of $P$ of size $\gtrsim n^{1-\delta}$ must contain isosceles triangles, and (2) all subsets of $P$ of size $\gtrsim n^{1/2-\delta}$ must contain repeated distances. These features give polynomially improved estimates for old problems of Erdős. The existence of a set satisfying property (1) confirms a conjecture of Erdős from 1980, whereas the existence of a set with property (2) answers a question of Conlon-Fox-Gasarch-Harris-Ulrich-Zbarsky in the negative.
Supports. An independent construction by different authors, inspired by the number-field counterexample but proceeding through a robust Ramanujan-type estimate; the counterexample follows by taking the whole set. Only the abstract was read on this pass; the proof itself was not checked here.
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- unknown
- Authors
- Lee, Sungchul, Pohoata, Cosmin, Zhu, Daniel G.
- Publisher
- arXiv.org
- Published
- Jul 6, 2026
- DOI
- None
- Link
- https://arxiv.org/abs/2607.05374
- Archived copy
- https://web.archive.org/web/20261011031210/https://arxiv.org/abs/2607.05374
- Stored text
- 4,740 characters, retrieved Sep 16, 2026 · 45 sections, 57 passages
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- Versions
- One known
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- Facts checked
- Nov 9, 2026
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