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The simplest inequality is that for sets $ F, G\\subset [0,N]$ there holds \\begin{equation*} N ^{-1} \\langle A_N \\mathbf 1_{F} , \\mathbf 1_{G} \\rangle \\ll \\frac{\\lvert F\\rvert \\cdot \\lvert G\\rvert} { N ^2 } \\Bigl( \\operatorname {Log} \\frac{\\lvert F\\rvert \\cdot \\lvert G\\rvert} { N ^2 } \\Bigr) ^{t}, \\end{equation*} where $ t=2$, or as"},"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":"2101.10401","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2021-01-25T20:36:23Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"bacc6ca8e72c8a5b01f0ada8663c80028fe085291b0c00a57ea8c2205ae28ead","abstract_canon_sha256":"ff2c2d1cf6a87fa0223c12492b9f9e1946220ba5ab479be8f01c8f6f436e8fae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:05:24.562477Z","signature_b64":"yovyF+NI+mSMph6DFIpZk7UPCo3WdwgyEcsTIKJW6hLcBVNV1mxpgJLqLlszgGKdjsasShVH7AIK9gWCnG4fAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e32e8dca82153d07412b36facb6b96817d503456a5aa29fbf12b4b5ab579462f","last_reissued_at":"2026-07-05T06:05:24.562029Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:05:24.562029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Endpoint $ \\ell ^{r}$ improving estimates for Prime averages","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Hamed Mousavi, Michael T. Lacey, Yaghoub Rahimi","submitted_at":"2021-01-25T20:36:23Z","abstract_excerpt":"Let $ \\Lambda $ denote von Mangoldt's function, and consider the averages \\begin{align*} A_N f (x) &=\\frac{1}{N}\\sum_{1\\leq n \\leq N}f(x-n)\\Lambda(n) . \\end{align*} We prove sharp $ \\ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, G\\subset [0,N]$ there holds \\begin{equation*} N ^{-1} \\langle A_N \\mathbf 1_{F} , \\mathbf 1_{G} \\rangle \\ll \\frac{\\lvert F\\rvert \\cdot \\lvert G\\rvert} { N ^2 } \\Bigl( \\operatorname {Log} \\frac{\\lvert F\\rvert \\cdot \\lvert G\\rvert} { N ^2 } \\Bigr) ^{t}, \\end{equation*} where $ t=2$, or as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.10401","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2101.10401/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2101.10401","created_at":"2026-07-05T06:05:24.562086+00:00"},{"alias_kind":"arxiv_version","alias_value":"2101.10401v2","created_at":"2026-07-05T06:05:24.562086+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.10401","created_at":"2026-07-05T06:05:24.562086+00:00"},{"alias_kind":"pith_short_12","alias_value":"4MXI3SUCCU6Q","created_at":"2026-07-05T06:05:24.562086+00:00"},{"alias_kind":"pith_short_16","alias_value":"4MXI3SUCCU6QOQJL","created_at":"2026-07-05T06:05:24.562086+00:00"},{"alias_kind":"pith_short_8","alias_value":"4MXI3SUC","created_at":"2026-07-05T06:05:24.562086+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.08893","citing_title":"Sharp $\\ell^p$-Improving Estimates for Fixed-Radius Discrete Spherical Averages","ref_index":21,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4MXI3SUCCU6QOQJLG35MW24WQF","json":"https://pith.science/pith/4MXI3SUCCU6QOQJLG35MW24WQF.json","graph_json":"https://pith.science/api/pith-number/4MXI3SUCCU6QOQJLG35MW24WQF/graph.json","events_json":"https://pith.science/api/pith-number/4MXI3SUCCU6QOQJLG35MW24WQF/events.json","paper":"https://pith.science/paper/4MXI3SUC"},"agent_actions":{"view_html":"https://pith.science/pith/4MXI3SUCCU6QOQJLG35MW24WQF","download_json":"https://pith.science/pith/4MXI3SUCCU6QOQJLG35MW24WQF.json","view_paper":"https://pith.science/paper/4MXI3SUC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2101.10401&json=true","fetch_graph":"https://pith.science/api/pith-number/4MXI3SUCCU6QOQJLG35MW24WQF/graph.json","fetch_events":"https://pith.science/api/pith-number/4MXI3SUCCU6QOQJLG35MW24WQF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4MXI3SUCCU6QOQJLG35MW24WQF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4MXI3SUCCU6QOQJLG35MW24WQF/action/storage_attestation","attest_author":"https://pith.science/pith/4MXI3SUCCU6QOQJLG35MW24WQF/action/author_attestation","sign_citation":"https://pith.science/pith/4MXI3SUCCU6QOQJLG35MW24WQF/action/citation_signature","submit_replication":"https://pith.science/pith/4MXI3SUCCU6QOQJLG35MW24WQF/action/replication_record"}},"created_at":"2026-07-05T06:05:24.562086+00:00","updated_at":"2026-07-05T06:05:24.562086+00:00"}