{"paper":{"title":"On restricted 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":"Dominique Maldague, Hong Wang, Larry Guth, Shaoming Guo, Shengwen Gan, Terence L. J. Harris","submitted_at":"2022-07-28T01:06:37Z","abstract_excerpt":"Let $\\gamma:[0,1]\\rightarrow \\mathbb{S}^{2}$ be a non-degenerate curve in $\\mathbb{R}^3$, that is to say, $\\det\\big(\\gamma(\\theta),\\gamma'(\\theta),\\gamma\"(\\theta)\\big)\\neq 0$. For each $\\theta\\in[0,1]$, let $V_\\theta=\\gamma(\\theta)^\\perp$ and let $\\pi_\\theta:\\mathbb{R}^3\\rightarrow V_\\theta$ be the orthogonal projections. We prove that if $A\\subset \\mathbb{R}^3$ is a Borel set, then for a.e. $\\theta\\in [0,1]$ we have $\\text{dim}(\\pi_\\theta(A))=\\min\\{2,\\text{dim} A\\}$. More generally, we prove an exceptional set estimate. For $A\\subset\\mathbb{R}^3$ and $0\\le s\\le 2$, define $E_s(A):=\\{\\theta\\in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.13844","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2207.13844/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}