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We prove the fixed-radius estimate $$\\|A_R f\\|_{\\ell^{p'}(\\mathbb{Z}^d)}\\leq C_{d,p,\\varepsilon}R^{-d(2/p-1)+\\varepsilon}\\|f\\|_{\\ell^p(\\mathbb{Z}^d)}$$ for $(d+2)/d\\leq p\\leq 2$, where $p'$ is the H\\\"older conjugate exponent of $p$ and $A_R$ is the probability average over the lattice sphere of radius $R$. This extends the fixed-radius estimates of Kesler--Lacey and Hughes to the sharp lower endpoint $p=(d+2)/d$."},"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":"2608.08893","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-08-09T20:14:04Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"cea2b52e8a4e44950c4e8f40172218e37c7a11bee67bf21d21948165adc0fbae","abstract_canon_sha256":"a3b8155ef65f0b0eb84d019d80d127cb078c23ea34a24828a557845909a620d0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T01:25:38.642153Z","signature_b64":"oWbVLEHM9G4Ep4lwPdj2oE3BMIIhXW8YA76RewqL2cSCZFRRTTk5l4xeqXlGvoceGqAye/sjMGLWgpSp9Jr2Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb699c6021fa1780d4d585f89298f7ea6e47bb6e9ad855bb2e0effe12cee40e9","last_reissued_at":"2026-08-11T01:25:38.639603Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T01:25:38.639603Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp $\\ell^p$-Improving Estimates for Fixed-Radius Discrete Spherical Averages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Fan Yang, Rui Han","submitted_at":"2026-08-09T20:14:04Z","abstract_excerpt":"Let $d\\geq 4$ and let $R>0$. When $d=4$, assume that $R^2\\in\\mathbb{N}\\setminus 4\\mathbb{N}$; when $d\\geq 5$, let $R^2\\in\\mathbb{N}$ be arbitrary. We prove the fixed-radius estimate $$\\|A_R f\\|_{\\ell^{p'}(\\mathbb{Z}^d)}\\leq C_{d,p,\\varepsilon}R^{-d(2/p-1)+\\varepsilon}\\|f\\|_{\\ell^p(\\mathbb{Z}^d)}$$ for $(d+2)/d\\leq p\\leq 2$, where $p'$ is the H\\\"older conjugate exponent of $p$ and $A_R$ is the probability average over the lattice sphere of radius $R$. This extends the fixed-radius estimates of Kesler--Lacey and Hughes to the sharp lower endpoint $p=(d+2)/d$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.08893","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.08893/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":"2608.08893","created_at":"2026-08-11T01:25:38.640460+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.08893v1","created_at":"2026-08-11T01:25:38.640460+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.08893","created_at":"2026-08-11T01:25:38.640460+00:00"},{"alias_kind":"pith_short_12","alias_value":"7NUZYYBB7ILY","created_at":"2026-08-11T01:25:38.640460+00:00"},{"alias_kind":"pith_short_16","alias_value":"7NUZYYBB7ILYBVGV","created_at":"2026-08-11T01:25:38.640460+00:00"},{"alias_kind":"pith_short_8","alias_value":"7NUZYYBB","created_at":"2026-08-11T01:25:38.640460+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/7NUZYYBB7ILYBVGVQX4JFGHX5J","json":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J.json","graph_json":"https://pith.science/api/pith-number/7NUZYYBB7ILYBVGVQX4JFGHX5J/graph.json","events_json":"https://pith.science/api/pith-number/7NUZYYBB7ILYBVGVQX4JFGHX5J/events.json","paper":"https://pith.science/paper/7NUZYYBB"},"agent_actions":{"view_html":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J","download_json":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J.json","view_paper":"https://pith.science/paper/7NUZYYBB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.08893&json=true","fetch_graph":"https://pith.science/api/pith-number/7NUZYYBB7ILYBVGVQX4JFGHX5J/graph.json","fetch_events":"https://pith.science/api/pith-number/7NUZYYBB7ILYBVGVQX4JFGHX5J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/action/storage_attestation","attest_author":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/action/author_attestation","sign_citation":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/action/citation_signature","submit_replication":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/action/replication_record"}},"created_at":"2026-08-11T01:25:38.640460+00:00","updated_at":"2026-08-11T01:25:38.640460+00:00"}