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When $f$ is isotropic or $n$ is at least $3$, we show that there is a $\\delta(f) \\in \\mathbb{Q} \\cap (0,1)$ such that $D_f(X) \\sim \\delta(f) X$ and call $\\delta(f)$ the density of $f$. We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set o"},"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":"2304.07399","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2023-04-14T21:28:27Z","cross_cats_sorted":[],"title_canon_sha256":"2a2f41e965a1b91ac669ee994b19f2875c1edd2ab8edd3b8bab377ab6385742b","abstract_canon_sha256":"758f6e0006061ee9d75c7765a52cd2f1571d8d2a5810a5c45735946900e5e6ea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:01:20.798281Z","signature_b64":"Ik4+3Sl/QAIh8llrGq+rMwVEL8gdAKFtAg1QUrcfleQ51yU96ZhQ1zzYsAkYjrWvKJlZiU9fQMOR5GWWv6DkBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"54a2c72e3888448b901999d1bce67982c284d94c33040ccfd782e0b7402952cd","last_reissued_at":"2026-07-05T06:01:20.797908Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:01:20.797908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Densities of integer sets represented by quadratic forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jeremy Rouse, Katherine Thompson, Paul Pollack, Pete L. 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