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I prove that the radial projections $\\pi_{x\\sharp}\\mu$ of $\\mu$ are absolutely continuous with respect to $\\mathcal{H}^{d - 1}$ for every centre $x \\in \\mathbb{R}^{d} \\setminus \\operatorname{spt} \\mu$, outside an exceptional set of dimension at most $2(d - 1) - s$. This is sharp. In fact, for $x$ outside an exceptional set as above, $\\pi_{x\\sharp}\\mu \\in L^{p}(S^{d - 1})$ for some $p > 1$."},"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":"1709.04653","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-09-14T08:02:26Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"b805bb061caa327575bb4f5e723f9008262f5334a320f2ee40b6a0bb42ec5122","abstract_canon_sha256":"b5bb6c5f517714dd5902713aaef8e4503caaeb4accf39bb8deeafc071441dde5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:11.018967Z","signature_b64":"MDMCNEAaCZySV25bt3A4VCl31IlRO5XHTRnMoZ4LlspjcOwqfivyZU6mGBeUxb7SKaZsxvtBkvYRvM3p3I8KAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ff8d9cbe62c025c4773565a70681f3d101a4b9c4496eecd05e8ceddcc4799c8","last_reissued_at":"2026-05-18T00:35:11.018499Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:11.018499Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the absolute continuity of radial projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Tuomas Orponen","submitted_at":"2017-09-14T08:02:26Z","abstract_excerpt":"Let $d \\geq 2$ and $d - 1 < s < d$. Let $\\mu$ be a compactly supported Radon measure in $\\mathbb{R}^{d}$ with finite $s$-energy. I prove that the radial projections $\\pi_{x\\sharp}\\mu$ of $\\mu$ are absolutely continuous with respect to $\\mathcal{H}^{d - 1}$ for every centre $x \\in \\mathbb{R}^{d} \\setminus \\operatorname{spt} \\mu$, outside an exceptional set of dimension at most $2(d - 1) - s$. This is sharp. 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