{"id":"openai-math/The-Falconer-distance-conjecture-in-all-dimensions","pdf":"https://github.com/openai/math/blob/main/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026","tags":["geometric-measure-theory","falconer-conjecture","distance-sets","harmonic-analysis"],"score":0,"title":"The Falconer distance conjecture in all dimensions","domain":"Real and complex analysis","family":"073","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CA","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://arxiv.org/abs/2610.22222","date":"2026-10-15","kind":"web","site":"arXiv.org","text":"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…","added":"2026-10-24","title":"A quantitative form of the planar Falconer distance theorem","author":"Guozheng Cheng, Xiang Fang, Menghan Li, Hongdou Qu et al.","record":"arxiv/2610.22222"}],"formalized":true,"family_title":"The Falconer distance conjecture"}