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The Falconer distance conjecture

For E⊂RdE\subset\mathbb R^d let Δ(E)={∣x−y∣:x,y∈E}\Delta(E)=\{|x-y|:x,y\in E\} be its distance set. Falconer (1985) showed that dim⁡HE>(d+1)/2\dim_H E>(d+1)/2 forces Δ(E)\Delta(E) to have positive Lebesgue measure, and lattice-type examples show the threshold cannot go below d/2d/2. Later thresholds were 4/34/3 in the plane (Wolff), d/2+1/3d/2+1/3 (Erdogan), 9/59/5 in R3\mathbb R^3 and further improvements via decoupling (Du, Guth, Ou, Wang, Wilson, Zhang; Du, Zhang), and 5/45/4 in the plane for pinned distances (Guth, Iosevich, Ou, Wang). Does every compact E⊂RdE\subset\mathbb R^d, d≥2d\ge2, with dim⁡HE>d/2\dim_H E>d/2 have L1(Δ(E))>0\mathcal L^1(\Delta(E))>0?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Geometric measure theory; harmonic analysis
Posed by
Kenneth Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1985)
Year posed
1985
Years open
41y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
60 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every integer d≥2d\ge2 and every compact E⊂RdE\subset\mathbb R^d with dim⁡HE>d/2\dim_H E>d/2, L1(Δ(E))>0\mathcal L^1(\Delta(E))>0. No regularity beyond the strict dimension bound is assumed. The paper does NOT treat the endpoint dim⁡HE=d/2\dim_H E=d/2, does NOT prove a pinned version (some yy with L1(Δy(E))>0\mathcal L^1(\Delta_y(E))>0) at the threshold d/2d/2. It states that it uses no decoupling theorem and no earlier distance theorem at an improved threshold; it does use the Orponen-Shmerkin incidence theorem.

What the AI did

The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 23, 2026. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source read against Falconer's question as the introduction states it; it is the full conjecture, every d≥2d\ge2, compact EE, strict threshold d/2d/2, with no packing-dimension, Ahlfors-regularity or pinned hypothesis. The proof was not refereed. Lean: the formalization catalogue lists only the planar case (PlanarFalconer, OAI.PlanarFalconer.mainTarget_proved). The all-dimensions statement is the release's Comparator challenge FalconerAllDimensions (OAI.Falconer.falconer_distance_conjecture), whose solution module OAI.MeasureTheory.Falconer.Campaign123PlanarFurstenbergProof exists at the pinned commit; that challenge is not in the formalization catalogue. Its statement was read here and states Theorem 1.1 exactly: for all d≥2d\ge2 and compact EE in EuclideanSpace with d/2<dim⁡HEd/2<\dim_H E, the distance set has positive volume. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The paper makes no endpoint claim and no single-pin claim at d/2d/2.

Sources

Changelog1 change

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