{"paper":{"title":"On the distance sets of AD-regular sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Tuomas Orponen","submitted_at":"2015-09-22T16:35:49Z","abstract_excerpt":"I prove that if $\\emptyset \\neq K \\subset \\mathbb{R}^{2}$ is a compact $s$-Ahlfors-David regular set with $s \\geq 1$, then $$\\dim_{\\mathrm{p}} D(K) = 1,$$ where $D(K) := \\{|x - y| : x,y \\in K\\}$ is the distance set of $K$, and $\\dim_{\\mathrm{p}}$ stands for packing dimension.\n  The same proof strategy applies to other problems of similar nature. For instance, one can show that if $\\emptyset \\neq K \\subset \\mathbb{R}^{2}$ is a compact $s$-Ahlfors-David regular set with $s \\geq 1$, then there exists a point $x_{0} \\in K$ such that $\\dim_{\\mathrm{p}} K \\cdot (K - x_{0}) = 1$. Specialising to prod"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06675","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}