The Falconer distance conjecture in all dimensions
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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A quantitative form of the planar Falconer distance theoremarXiv.orgGuozheng Cheng, Xiang Fang, Menghan Li, Hongdou Qu et al.Oct 15, 2026in the index
We give a quantitative planar form of a multiscale argument for Falconer's distance problem. For every $0 1/\varepsilon$. In particular, compact planar sets of Hausdorff dimension greater than one have distance sets of positive Lebesgue measure. The proof develops the multiscale method of the…