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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2207.13844","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2022-07-28T01:06:37Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"6943ea01b902b795a559118af32bba5a02905fa74cd36444369be548d734aa37","abstract_canon_sha256":"cb273106e7eb28382d96ef5434d1021f1e7d4df80730843cd96f949966d42127"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:00:28.330976Z","signature_b64":"GzQQrQBfYIwhwGvkrmJg8hefFj3PNPTncqqucGBHlaym5YG2ypj89ez6PFMWt11vby8LVtE1w1gcEgeRCNkIAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1756303afa670f0a4111e8aa18d574de752bf5a59d2d28216b62f408b13cd730","last_reissued_at":"2026-07-05T08:00:28.330489Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:00:28.330489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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