{"paper":{"title":"Improved bounds for restricted families of projections to planes in $\\mathbb{R}^{3}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Laura Venieri, Tuomas Orponen","submitted_at":"2017-11-24T11:57:50Z","abstract_excerpt":"For $e \\in S^{2}$, the unit sphere in $\\mathbb{R}^3$, let $\\pi_{e}$ be the orthogonal projection to $e^{\\perp} \\subset \\mathbb{R}^{3}$, and let $W \\subset \\mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. We prove that if $K \\subset \\mathbb{R}^{3}$ is a Borel set with $\\dim_{\\mathrm{H}} K \\leq \\tfrac{3}{2}$, then $\\dim_{\\mathrm{H}} \\pi_{e}(K) = \\dim_{\\mathrm{H}} K$ for $\\mathcal{H}^{1}$ almost every $e \\in S^{2} \\cap W$, where $\\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure and $\\dim_{\\mathrm{H}}$ the Hausdorff dimension. This was known earlier, due to J\\\"arvenp\\\"a\\\"a,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.08934","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"}