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The Falconer distance conjecture

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.

Manuscripts in this family · 1

Sep 23, 2026·paper.pdfLean

We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝ

^{d}

with Hausdorff dimension greater than

d/2

determines a set of Euclidean distances of positive Lebesgue measure.

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