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Under mild hypothesis, Corvaja and Zannier proved that $\\mathcal{N}$ has zero asymptotic density. We prove that $\\#(\\mathcal{N} \\cap [1, x]) \\ll x \\cdot (\\log\\log x / \\log x)^h$ for all $x \\geq 3$, where $h$ is a positive integer that can be computed in terms of $F$ and $G$. Assuming the Hardy-Littlewood $k$-tuple conjecture, 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":"1703.10047","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-03-29T14:07:39Z","cross_cats_sorted":[],"title_canon_sha256":"2c9dbca9d1268b446a5f885806e200756654726c9d99711c4ad5e18929783c0d","abstract_canon_sha256":"63f3d817a2393efde38a7eb1e3d31ed5020bc72bf1d8a5b725a2c3891fc7084e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:38.769288Z","signature_b64":"JpbrxeJICV05RZwjVUO+GuDUdf9X9jkekZYVY0/efAWoTMH5Nj7N1pP+DwQUj9V3TskaW5cesm1G7NMR2Gc6Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c72e07bb12800983014f8a5af4b138a84a7efa2729c168a4e25d82d2e99aeb6","last_reissued_at":"2026-05-18T00:36:38.768670Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:38.768670Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distribution of integral values for the ratio of two linear recurrences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carlo Sanna","submitted_at":"2017-03-29T14:07:39Z","abstract_excerpt":"Let $F$ and $G$ be linear recurrences over a number field $\\mathbb{K}$, and let $\\mathfrak{R}$ be a finitely generated subring of $\\mathbb{K}$. Furthermore, let $\\mathcal{N}$ be the set of positive integers $n$ such that $G(n) \\neq 0$ and $F(n) / G(n) \\in \\mathfrak{R}$. Under mild hypothesis, Corvaja and Zannier proved that $\\mathcal{N}$ has zero asymptotic density. We prove that $\\#(\\mathcal{N} \\cap [1, x]) \\ll x \\cdot (\\log\\log x / \\log x)^h$ for all $x \\geq 3$, where $h$ is a positive integer that can be computed in terms of $F$ and $G$. 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