{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ZJH63FIA2RZMNHI25H4HU56GVS","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"49c1588184c5b3ddca56b2c08c912483f2eeefc67a49b85f1723485bcbacc885","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-09-06T16:44:27Z","title_canon_sha256":"5787d32cc1664a41a094a0263f44b25967f1274211bcb040cd339e96ecbfbecb"},"schema_version":"1.0","source":{"id":"1909.02883","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1909.02883","created_at":"2026-07-05T01:11:41Z"},{"alias_kind":"arxiv_version","alias_value":"1909.02883v2","created_at":"2026-07-05T01:11:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.02883","created_at":"2026-07-05T01:11:41Z"},{"alias_kind":"pith_short_12","alias_value":"ZJH63FIA2RZM","created_at":"2026-07-05T01:11:41Z"},{"alias_kind":"pith_short_16","alias_value":"ZJH63FIA2RZMNHI2","created_at":"2026-07-05T01:11:41Z"},{"alias_kind":"pith_short_8","alias_value":"ZJH63FIA","created_at":"2026-07-05T01:11:41Z"}],"graph_snapshots":[{"event_id":"sha256:48bad37451661fcf534628872ef53ed298ac534e7977effca158b7d30ccfed4b","target":"graph","created_at":"2026-07-05T01:11:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1909.02883/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Consider averages along the prime integers $ \\mathbb P $ given by \\begin{equation*} \\mathcal{A}_N f (x) = N ^{-1} \\sum_{ p \\in \\mathbb P \\;:\\; p\\leq N} (\\log p) f (x-p). \\end{equation*} These averages satisfy a uniform scale-free $ \\ell ^{p}$-improving estimate. For all $ 1< p < 2$, there is a constant $ C_p$ so that for all integer $ N$ and functions $ f$ supported on $ [0,N]$, there holds \\begin{equation*} N ^{-1/p' }\\lVert \\mathcal{A}_N f\\rVert_{\\ell^{p'}} \\leq C_p N ^{- 1/p} \\lVert f\\rVert_{\\ell^p}. \\end{equation*} The maximal function $ \\mathcal{A}^{\\ast} f =\\sup_{N} \\lvert \\mathcal{A}_N ","authors_text":"Ben Krause, Fan Yang, Michael Lacey, Rui Han","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-09-06T16:44:27Z","title":"Averages Along the Primes: Improving and Sparse Bounds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.02883","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5b87823f738aea50eb68844f9f9ced384562bed658ba373b291eb9a151c22397","target":"record","created_at":"2026-07-05T01:11:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"49c1588184c5b3ddca56b2c08c912483f2eeefc67a49b85f1723485bcbacc885","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-09-06T16:44:27Z","title_canon_sha256":"5787d32cc1664a41a094a0263f44b25967f1274211bcb040cd339e96ecbfbecb"},"schema_version":"1.0","source":{"id":"1909.02883","kind":"arxiv","version":2}},"canonical_sha256":"ca4fed9500d472c69d1ae9f87a77c6ac85ec646684e31370e6adfeba2af75a54","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ca4fed9500d472c69d1ae9f87a77c6ac85ec646684e31370e6adfeba2af75a54","first_computed_at":"2026-07-05T01:11:41.525765Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:11:41.525765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"c+WTcEC5D7fNU2L8iJSnpSWyIvmBnRrr+pMe69iCoVecBbYqsD61qo14SBnvXjiJIoj4H0pLXJ9wghzNkOKJAw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:11:41.526354Z","signed_message":"canonical_sha256_bytes"},"source_id":"1909.02883","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5b87823f738aea50eb68844f9f9ced384562bed658ba373b291eb9a151c22397","sha256:48bad37451661fcf534628872ef53ed298ac534e7977effca158b7d30ccfed4b"],"state_sha256":"136913dcd667d7715fbbe72823c994855b852f59ce606f30d096ffaad94c84d4"}