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Math-2441 · mathematics · Geometry

Minimum Distinct Distances Among Planar Point Sets

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Determine the minimum number $f(n)$ of distinct distances determined by $n$ distinct points in the Euclidean plane.

Formulation references: [S1], [S2].

Short English statement

Determine the minimum number $f(n)$ of distinct distances determined by $n$ distinct points in the Euclidean plane.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2441-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2441

For every fixed $\varepsilon>0$, the claimed almost-all-pins theorem implies at least $n^{1-\varepsilon}$ distinct distances in every sufficiently large planar $n$-point set. It improves a lower bound for the quantity asked about, while leaving its exact value and sharp asymptotic order undetermined.

Minimum Distinct Distances Among Planar Point Sets

OpenAI's claim

Almost all planar pins: For any fixed positive $\varepsilon$, the claimed number of pins determining fewer than $n^{1-\varepsilon}$ nonzero distances is $o(n)$ uniformly over planar $n$-point sets. This is an almost-all-pins conclusion rather than an exact linear lower bound at every pin.

Mathematical statement of the claimed result

Mathematical claim 1.
For finite $P\subset\mathbb R^2$ put $D_x(P)=\{|y-x|:y\in P\setminus\{x\}\}$ and $k_P(x,y)=|\{z\in P\setminus\{x\}:|z-x|=|y-x|\}|$. The claim states that for every $s>0$, \[\sup_{|P|=n}\frac{|\{(x,y)\in P^2:x\ne y,\ k_P(x,y)\ge n^s\}|}{n(n-1)}\longrightarrow0.\] Consequently, for every $\epsilon>0$, $\sup_{|P|=n}|\{x:|D_x(P)|<n^{1-\epsilon}\}|/n\to0$. Thus all but $o(n)$ pins determine at least $n^{1-\epsilon}$ distances, uniformly over the configuration; the sharp $n/\sqrt{\log n}$ global scale is not asserted.

Outline of the argument

Almost all planar pins: An extremal pair graph is extracted from large distance fibers while retaining nonconcentration within those fibers. Real-closed-field transfer moves finite configurations into number fields without altering their distance relations. Factorization into two complex coordinates and the product formula balance local closeness partitions across absolute values, giving an incompatibility with the assumed abundance of large fibers.

Formal verification

Family 167 describes Lean coverage for the weak pinned-distance conclusion and the separate fixed-power unit-distance improvement.

Almost all planar pins: The manuscript links PinnedDistances.lean (declaration OAI.WeakPinned.main), PlanarUnitDistances.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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