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We develop a general framework to link $\\pi^x(E)$, $x\\in F$ and $\\pi^y(F)$, $y\\in E$, for any Borel set $F\\subset\\mathbb{R}^d$. In particular, whether $\\dim_{\\mathcal"},"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":"1903.12093","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-03-28T16:15:28Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"f0c38b0b07e8fd08aad1b53382e2c1bbdbd468d68436f8131421ed8c8c5019ce","abstract_canon_sha256":"46ab2d7c4938d609961f580ed415d6f325606513a039679f1d4f6b032306c11c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:32.636704Z","signature_b64":"EbC3F9/SSVOJfM7ADLetEMHZg2LbZr0QjVS4Eneds9FYbhkb3tmAVkBm0bZG8L+TbgrJtAoXPHlO+GJCdwJ0DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d3217b3396a02ebe53e389061ca201a9268b159a650e1cfd15315c615cca194","last_reissued_at":"2026-05-17T23:49:32.636116Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:32.636116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Hausdorff dimension of radial projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Bochen Liu","submitted_at":"2019-03-28T16:15:28Z","abstract_excerpt":"For any $x\\in\\mathbb{R}^d$, $d\\geq 2$, denote $\\pi^x: \\mathbb{R}^d\\backslash\\{x\\}\\rightarrow S^{d-1}$ as the radial projection $$\\pi^x(y)=\\frac{y-x}{|y-x|}. $$\n  Given a Borel set $E\\subset{\\Bbb R}^d$, $\\dim_{\\mathcal{H}} E\\leq d-1$, in this paper we investigate for how many $x\\in \\mathbb{R}^d$ the radial projection $\\pi^x$ preserves the Hausdorff dimension of $E$, namely whether $\\dim_{\\mathcal{H}}\\pi^x(E)=\\dim_{\\mathcal{H}} E$. We develop a general framework to link $\\pi^x(E)$, $x\\in F$ and $\\pi^y(F)$, $y\\in E$, for any Borel set $F\\subset\\mathbb{R}^d$. 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