Result 167, Combinatorics

Planar distinct distances and unit-distance bounds

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 $n^{1-\varepsilon}$ distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by $O(n^{4/3-\delta})$ for an absolute δ > 0.

Lean formalization Proof

The bigger picture

Why it matters

How many different distances must a cloud of points contain, and how often can one distance repeat? These manuscripts claim strong restrictions on both kinds of behavior for arbitrary point sets in the plane.

What changes?

The manuscripts report that, in any n-point Euclidean planar set, all but o(n) points determine at least n to the power 1 minus epsilon distinct nonzero distances to other points, for each fixed positive epsilon. The exceptional fraction tends to zero as n grows with epsilon fixed. Separately, unordered pairs exactly one unit apart number at most a constant times n to the power 4/3 minus delta, for some absolute positive delta.

What does that help mathematicians do?

The first claim controls distances viewed from almost every individual point, not merely distances collected across the whole set. Equivalently, centering circles at almost any point requires nearly a linear number of distinct radii, in the stated power-law sense, to reach all other points. This rules out widespread concentration on few radii, while leaving exceptional points possible. The second claim independently limits repetitions of one fixed distance.

Are there practical applications?

The immediate value is foundational, connecting planar geometry with combinatorial counting. For example, joining pairs exactly one unit apart produces a graph whose edges would obey the claimed improved bound. That gives researchers a necessary constraint on which graphs can arise from such planar configurations, rather than a practical construction algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The weak pinned planar distance theorem

September 23, 2026 27 pages Main result formalized in Lean

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 $n^{1-\varepsilon}$ distinct nonzero distances.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-weak-pinned-planar-distance-theorem-September-23-2026,
  author = {{OpenAI}},
  title = {{The weak pinned planar distance theorem}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf}{OAI:The-weak-pinned-planar-distance-theorem-September-23-2026}},
  year = {2026}
}

A power saving for planar unit distances

September 23, 2026 53 pages

We prove a power saving for the planar unit-distance problem: for some absolute $\beta\lt 4/3$ , every set of n points in the Euclidean plane determines $O(n^\beta)$ unordered pairs at unit distance.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-power-saving-for-planar-unit-distances-September-23-2026,
  author = {{OpenAI}},
  title = {{A power saving for planar unit distances}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf}{OAI:A-power-saving-for-planar-unit-distances-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/167.md.

Planar distinct distances and unit-distance bounds

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result shows that large repeated distance fibers are rare in arbitrary finite planar point sets. For each fixed $s>0$ , the largest possible fraction of ordered distinct pairs $(x,y)$ whose distance from the pin $x$ occurs at least $n^s$ times tends to zero as the set size $n$ grows. Consequently, for every $\varepsilon>0$ , the fraction of pins determining fewer than $n^{1-\varepsilon}$ distances tends uniformly to zero. The separate unit-distance power-saving theorem is not included.

The planar unit-distance problem asks how many pairs at distance one can occur among $n$ points. The formalization proves that there are absolute constants $C>0$ and $1\le\beta<4/3$ such that every finite planar point set of size $n$ determines at most $Cn^\beta$ unordered unit-distance pairs. The same constants work for every $n$ , giving a fixed power improvement over the classical exponent $4/3$ .

Comparator links

Result	Comparator statement
Weak pinned planar distance theorem	PinnedDistances.lean
Power saving for planar unit distances	PlanarUnitDistances.lean

At a glance

Subject: Combinatorics
Manuscripts: 2
Pages: 80
Dated: Sep 23, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Weak pinned Erdős distinct-distance conjectureProved
Planar unit-distance problemPartial progress

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

Most similar results

166, CombinatoricsThe higher-dimensional Erdős distinct-distances conjecture
183, CombinatoricsPower savings for planar halving lines and k-sets
191, CombinatoricsA power improvement in the Heilbronn triangle lower bound
089, Convex and metric geometryBounded-distortion L₁ embeddings of planar and bounded-treewidth graphs
097, Convex and metric geometryThe Euclidean Steinitz–Bergström bound
099, Convex and metric geometryThe sharp exponential scale of edit-distance distortion

By similarity of summaries and abstracts.

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