{"paper":{"title":"A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"A. Greenleaf, A. Iosevich, B. Liu, E. Palsson","submitted_at":"2013-06-15T18:44:05Z","abstract_excerpt":"We obtain nontrivial exponents for Erd\\H os-Falconer type problems. Let $T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes determined by $(k+1)$-tuples of points from $E$. We prove that there exists $s_0(d)<d$ such that, if $E \\subset {\\Bbb R}^d,\\, d \\ge 2$, with $dim_{{\\mathcal H}}(E)>s_0(d)$, then the ${k+1 \\choose 2}$-dimensional Lebesgue measure of $T_k(E)$ is positive. Results were previously obtained for triangles in the plane \\cite{GI12} and in higher dimensions \\cite{GGIP12}. In this paper, we improve upon those exponents, using a group-theoretic method that sheds "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3598","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"}