∑math20

Real and complex analysis · Result 073

The Falconer distance conjecture

1 paperLean

Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set \(E\subset\mathbb R^d\) with Hausdorff dimension greater than \(d/2\) determines a set of Euclidean distances of positive Lebesgue measure.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 1

First page of The Falconer distance conjecture in all dimensions

The Falconer distance conjecture in all dimensions

OpenAI · September 23, 2026

We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝd with Hausdorff dimension greater than \(d/2\) determines a set of Euclidean distances of positive Lebesgue measure.

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

What was formalized in Lean

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

Scope

The formalization proves the Falconer distance conjecture in every dimension. For every integer \(d\ge2\) and compact set \(E\subset\mathbb R^d\) with Hausdorff dimension greater than \(d/2\), the set \(\{\lVert x-y\rVert:x,y\in E\}\) has positive Lebesgue measure. The linked statements include both this all-dimensional result and the earlier planar case.

Comparator links

ResultComparator statement
Planar Falconer distance theoremPlanarFalconer.lean
Falconer distance theorem in every dimensionFalconerAllDimensions.lean

Scope note on GitHub →