{"paper":{"title":"On three point configurations determined by subsets of the Euclidean plane, the associated bilinear operator and applications to discrete geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Allan Greenleaf","submitted_at":"2010-09-13T18:27:38Z","abstract_excerpt":"We prove that if the Hausdorff dimension of a compact set $E \\subset {\\Bbb R}^2$ is greater than 7/4, then the set of {\\ag three-point configurations determined by $E$ has positive three-dimensional measure}. We establish this by showing that {\\ag a} natural measure on the set of {\\ag such configurations} has {\\ag Radon-Nikodym derivative} in $L^{\\infty}$ if $\\dH(E)> 7/4$, and the index 7/4 in this last result cannot, in general, be improved. This problem naturally leads to the study of a bilinear convolution operator, $$ B(f,g)(x)=\\int \\int f(x-u) g(x-v)\\, dK(u,v),$$ where $K$ is surface meas"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2471","kind":"arxiv","version":4},"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"}