{"paper":{"title":"On the Mattila-Sjolin theorem for distance sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Krystal Taylor, Mihalis Mourgoglou","submitted_at":"2011-10-31T14:31:14Z","abstract_excerpt":"We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \\subset {\\Bbb R}^d$, $d \\ge 2$, is greater than $\\frac{d+1}{2}$, then the distance set $\\Delta(E)=\\{|x-y|: x,y \\in E \\}$ contains an interval. We prove this result for distance sets $\\Delta_B(E)=\\{{||x-y||}_B: x,y \\in E \\}$, where ${|| \\cdot ||}_B$ is the metric induced by the norm defined by a symmetric bounded convex body $B$ with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6805","kind":"arxiv","version":1},"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"}