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Math-2626 · mathematics · Combinatorics

Power improvement over the planar unit-distance exponent four thirds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Do absolute constants $\epsilon >0$ and $C>0$ exist such that every n-point set in $R^2$ has at most C $n^{4/3-\epsilon }$ unordered pairs at Euclidean distance one?

Short English statement

Do absolute constants $\epsilon >0$ and $C>0$ exist such that every n-point set in $R^2$ has at most C $n^{4/3-\epsilon }$ unordered pairs at Euclidean distance one?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2626-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2626

Planar unit-distance power saving: The main theorem gives absolute C and 1<=beta<4/3 with u(n)<=C n^beta for every n>=0. Setting epsilon=4/3-beta gives the fixed saving for all finite planar sets, counting unordered pairs and with no general-position or rational-direction hypothesis.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Power improvement over the planar unit-distance exponent four thirds

OpenAI's claim

Planar unit-distance power saving: The unit-distance manuscript claims an absolute exponent $\beta<4/3$ bounding the number of unordered unit pairs by $C n^\beta$. The point configuration is otherwise arbitrary.

Mathematical statement of the claimed result

Mathematical claim 1.
For $P\subset\mathbb R^2$ of $n$ distinct points let $u(P)$ count unordered pairs at Euclidean distance $1$, and let $u(n)=\max_{|P|=n}u(P)$. The claim is that absolute constants $C<\infty$ and $1\le\beta<4/3$ satisfy \[u(n)\le Cn^\beta\quad(n\ge0).\] The unspecified saving $4/3-\beta>0$ improves the known exponent but does not establish every proposed near-linear bound.

Outline of the argument

Planar unit-distance power saving: A near-extremal incidence sequence is transferred to algebraic configurations and analyzed using distance identities over a negligible-height coefficient field. After differentiating those identities, valuations distinguish the individual radicands and allow independent square-root sign changes. One unused sign change contradicts the differentiated system; a separate exclusion of popular unit circles handles the possible square-class failure, preventing approach to exponent $4/3$.

Formal verification

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

Planar unit-distance power saving: 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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