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Let $\\gamma(A) = \\sup \\{r : A \\hbox{ is coarsely computable at density } r\\}$. We study the interactions of these concepts with Turing reducibility. For example, we show that if $r \\in (0,1]$ there are sets $A_0, A_1$ such that $\\gamma(A_0) = \\gamma(A_1) = r$ where $A_0$ is coarsely computable at density $r$ while $A_1$ is not coarsely computable at density $r$. We show that a real $r \\in [0,1]$ is"},"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":"1505.01901","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-05-08T01:15:16Z","cross_cats_sorted":[],"title_canon_sha256":"4d04e6750d8caa68473a4457c3939e7cfc29eaf48e6a76cdf1b689f92e6350ea","abstract_canon_sha256":"67c936d676b81ae282118f9d649398ae544d91fd33084d268615177f52fdbbf6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:38.572909Z","signature_b64":"Ruzz2rFJ4tz23nRXDxnUwg4uqqzTKxs/AY3v51slILJDKKuylV9k8RvDkHo47dI9cZpa4HUByLWQXELReaJgCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f81976558ce9e4f545e8d1c99d32abcf5a33f81c19c5a08596645c104912cbca","last_reissued_at":"2026-05-18T02:16:38.572272Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:38.572272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic density and the coarse computability bound","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Carl G. Jockusch, Denis R. Hirschfeldt, Jr., Paul E. Schupp, Timothy H. McNicholl","submitted_at":"2015-05-08T01:15:16Z","abstract_excerpt":"For $r \\in [0,1]$ we say that a set $A \\subseteq \\omega$ is \\emph{coarsely computable at density} $r$ if there is a computable set $C$ such that $\\{n : C(n) = A(n)\\}$ has lower density at least $r$. Let $\\gamma(A) = \\sup \\{r : A \\hbox{ is coarsely computable at density } r\\}$. We study the interactions of these concepts with Turing reducibility. For example, we show that if $r \\in (0,1]$ there are sets $A_0, A_1$ such that $\\gamma(A_0) = \\gamma(A_1) = r$ where $A_0$ is coarsely computable at density $r$ while $A_1$ is not coarsely computable at density $r$. We show that a real $r \\in [0,1]$ is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01901","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1505.01901","created_at":"2026-05-18T02:16:38.572378+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.01901v1","created_at":"2026-05-18T02:16:38.572378+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01901","created_at":"2026-05-18T02:16:38.572378+00:00"},{"alias_kind":"pith_short_12","alias_value":"7AMXMVMM5HSP","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_16","alias_value":"7AMXMVMM5HSPKRPI","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_8","alias_value":"7AMXMVMM","created_at":"2026-05-18T12:29:07.941421+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7AMXMVMM5HSPKRPI2HEZ2MVLZ5","json":"https://pith.science/pith/7AMXMVMM5HSPKRPI2HEZ2MVLZ5.json","graph_json":"https://pith.science/api/pith-number/7AMXMVMM5HSPKRPI2HEZ2MVLZ5/graph.json","events_json":"https://pith.science/api/pith-number/7AMXMVMM5HSPKRPI2HEZ2MVLZ5/events.json","paper":"https://pith.science/paper/7AMXMVMM"},"agent_actions":{"view_html":"https://pith.science/pith/7AMXMVMM5HSPKRPI2HEZ2MVLZ5","download_json":"https://pith.science/pith/7AMXMVMM5HSPKRPI2HEZ2MVLZ5.json","view_paper":"https://pith.science/paper/7AMXMVMM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.01901&json=true","fetch_graph":"https://pith.science/api/pith-number/7AMXMVMM5HSPKRPI2HEZ2MVLZ5/graph.json","fetch_events":"https://pith.science/api/pith-number/7AMXMVMM5HSPKRPI2HEZ2MVLZ5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7AMXMVMM5HSPKRPI2HEZ2MVLZ5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7AMXMVMM5HSPKRPI2HEZ2MVLZ5/action/storage_attestation","attest_author":"https://pith.science/pith/7AMXMVMM5HSPKRPI2HEZ2MVLZ5/action/author_attestation","sign_citation":"https://pith.science/pith/7AMXMVMM5HSPKRPI2HEZ2MVLZ5/action/citation_signature","submit_replication":"https://pith.science/pith/7AMXMVMM5HSPKRPI2HEZ2MVLZ5/action/replication_record"}},"created_at":"2026-05-18T02:16:38.572378+00:00","updated_at":"2026-05-18T02:16:38.572378+00:00"}