{"paper":{"title":"On sets of directions determined by subsets of ${\\Bbb R}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Mihalis Mourgoglou, Steven Senger","submitted_at":"2010-09-21T18:24:13Z","abstract_excerpt":"Given $E \\subset \\mathbb{R}^d$, $d \\ge 2$, define ${\\mathcal D}(E) \\equiv {(x-y)/|x-y|: x,y \\in E} \\subset S^{d-1},$ the set of directions determined by $E$. We prove that if the Hausdorff dimension of $E$ is greater than $d-1$, then $\\sigma({\\mathcal D}(E))>0$, where $\\sigma$ denotes the surface measure on $S^{d-1}$. This result is sharp since the conclusion fails to hold if $E$ is a $(d-1)$-dimensional hyper-plane. This result can be viewed as a continuous analog of a recent result of Pach, Pinchasi, and Sharir (\\cite{PPS04}, \\cite{PPS07}) on angles determined by finite subsets of $\\mathbb{R"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4169","kind":"arxiv","version":2},"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"}