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The Falconer distance conjecture

Theorem 1.1: for every integer $d\ge2$ and every compact $E\subset\mathbb R^d$ with $\dim_H E>d/2$ , $\mathcal L^1(\Delta(E))>0$ . No regularity beyond the strict dimension bound is assumed. The paper does NOT treat the endpoint $\dim_H E=d/2$ , does NOT prove a pinned version (some $y$ with $\mathcal L^1(\Delta_y(E))>0$ ) at the threshold $d/2$ . 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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Falconer distance conjecture · Falconer distance conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The Falconer distance conjecture

specialist registry · passed

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 $d\ge2$ , compact $E$ , strict threshold $d/2$ , 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 $d\ge2$ and compact $E$ in EuclideanSpace with $d/2<\dim_H E$ , 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 $d/2$ .