{"paper":{"title":"An elementary approach to simplexes in thin subsets of Euclidean space","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Allan Greenleaf, Bochen Liu, Eyvindur Palsson","submitted_at":"2016-08-16T20:53:27Z","abstract_excerpt":"We prove that if the Hausdorff dimension of $E \\subset {\\Bbb R}^d$, $d \\ge 3$, is greater than $\\min \\left\\{ \\frac{dk+1}{k+1}, \\frac{d+k}{2} \\right\\},$ then the ${k+1 \\choose 2}$-dimensional Lebesgue measure of $T_k(E)$, the set of congruence classes of $k$-dimensional simplexes with vertices in $E$, is positive. This improves the best bounds previously known, decreasing the $\\frac{d+k+1}{2}$ threshold obtained in Erdo\\u{g}an-Hart-Iosevich (2012) to $\\frac{d+k}{2}$ via a different and conceptually simpler method. We also give a simpler proof of the $d-\\frac{d-1}{2d}$ threshold for $d$-dimensio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04777","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"}