{"paper":{"title":"On volumes determined by subsets of Euclidean space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Allan Greenleaf, Mihalis Mourgoglou","submitted_at":"2011-10-31T13:39:21Z","abstract_excerpt":"Given $E \\subset {\\Bbb R}^d$, define the \\emph{volume set} of $E$, ${\\mathcal V}(E)= \\{det(x^1, x^2, ... x^d): x^j \\in E\\}$. In $\\R^3$, we prove that ${\\mathcal V}(E)$ has positive Lebesgue measure if either the Hausdorff dimension of $E\\subset \\Bbb R^3$ is greater than 13/5, or $E$ is a product set of the form $E=B_1\\times B_2\\times B_3$ with $B_j\\subset\\R,\\, dim_{\\mathcal H}(B_j)>2/3,\\, j=1,2,3$. We show that the same conclusion holds for $\\V(E)$ of Salem subsets $E\\subset\\R^d$ with $\\hde>d-1$, and give applications to discrete combinatorial geometry."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6790","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"}