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Math-2442-A1 · revision r1 · on Math-2442

OpenAI claim summary for Math-2442

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Falconer distances: The manuscript’s result proves the compact-set assertion in every dimension, including d=2. For a Borel planar set of dimension >1, the compact-subset characterization of Borel Hausdorff dimension supplies a compact subset of dimension >1, and its positive-measure distance set lies in the original distance set.

The claimed result implies the stated question and also supplies the additional strength described above.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Falconer Distance Conjecture in the Plane
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Falconer Distance Conjecture in the Plane

OpenAI's claim

Falconer distances: For every $d\ge2$, the manuscript claims positive one-dimensional measure for the distance set of a compact subset of $\mathbb R^d$ with dimension exceeding $d/2$. Both the ambient dimension and the set itself are unrestricted within these hypotheses.

Mathematical statement of the claimed result

Mathematical claim 1.
For every integer $d\ge2$ and compact $E\subset\mathbb R^d$, \[\dim_H E>d/2\quad\Longrightarrow\quad\mathcal L^1\left(\{|x-y|:x,y\in E\}\right)>0.\] Here $\dim_H$ is Hausdorff dimension and $\mathcal L^1$ is Lebesgue measure on the distance axis.

Outline of the argument

Falconer distances: Auxiliary projections first provide angular nonconcentration and filters on pairs of points. A fractal Fourier estimate and a graph estimate with separate angular variables control shell approximations after small additional deletions. The filters are brought back to distances of the original pairs, and monotone shell summation produces a nonzero absolutely continuous distance measure.

Formal verification

Family 073 describes Lean coverage for the positive-measure distance conclusion above the half-dimensional threshold.

Falconer distances: The manuscript links FalconerAllDimensions.lean, PlanarFalconer.lean (declaration OAI.PlanarFalconer.mainTarget_proved).

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