Result 073, Real and complex analysis

The Falconer distance conjecture

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

Lean formalization Proof

The bigger picture

Why it matters

A set of points can have a complicated fractal shape, yet the distances between its points may collectively occupy positive length. The manuscript claims a dimension threshold guaranteeing this in every Euclidean dimension from two upward.

What changes?

The manuscript reports a resolution of the Falconer distance conjecture for every integer dimension d at least two. Its claim applies to every compact set E, meaning a closed, bounded set, whose Hausdorff dimension exceeds d/2. Hausdorff dimension measures size at fine scales and can be fractional. The conclusion is that the set of all Euclidean distances between pairs of points in E has positive Lebesgue measure, the usual measure of length on the line.

What does that help mathematicians do?

The claimed result would let researchers infer a substantial collection of distances from dimension alone, without further assumptions about the compact set's shape. Conversely, a compact set whose distances have zero total length could not have Hausdorff dimension above half the ambient dimension. The strict threshold matters: the claim says nothing at equality, and positive measure does not by itself guarantee an interval of distances.

Are there practical applications?

The immediate value is foundational: the claim links the fine-scale size of a geometric set to the length occupied by its pairwise distances. It would provide a general implication for studying distance sets, rather than a procedure for calculating them. The supplied abstract describes no practical implementation or computational guarantee.

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.

Manuscript

The Falconer distance conjecture in all dimensions

September 23, 2026 64 pages Main result formalized in Lean

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/2d/2 determines a set of Euclidean distances of positive Lebesgue measure.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/073.md.

The Falconer distance conjecture

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≥2d\ge2 and compact set E⊂RdE\subset\mathbb R^d with Hausdorff dimension greater than d/2d/2, the set {∥x−y∥:x,y∈E}\{\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

Result Comparator statement
Planar Falconer distance theorem PlanarFalconer.lean
Falconer distance theorem in every dimension FalconerAllDimensions.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.