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As a consequence we also obtain pointwise ergodic theorem along the set $\\mathbf{N}_{h}$."},"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":"1305.0575","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-05-02T20:38:30Z","cross_cats_sorted":[],"title_canon_sha256":"c4137d4e4c84848e434a9d342319b123a18df0d9bed1cd9c8b3dbe5ff2e340a3","abstract_canon_sha256":"25a8cf6ab535b1edf13e90ec8f6c42e83ddac8e06bd69356ea9f36564b58c03d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:30.790900Z","signature_b64":"KQm1XQGP07q4pQSGqZaS4e47ERwm6pscJ/o28xUcCYLDrrgOIyoHDkfGgnro9ifY2DmbrTGYd0sJz6X2Q8cQDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1297001c594ab203c5983045f3f454d871d277369a874e8f3fffabcd64902dab","last_reissued_at":"2026-05-18T02:54:30.790438Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:30.790438Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak type (1, 1) inequalities for discrete rough maximal functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Mariusz Mirek","submitted_at":"2013-05-02T20:38:30Z","abstract_excerpt":"The aim of this paper is to show that the discrete maximal function $$\\mathcal{M}_{h}f(x)=\\sup_{N\\in\\mathbb{N}}\\frac{1}{|\\mathbf{N}_{h}\\cap[1, N]|}\\Big|\\sum_{n\\in \\mathbf{N}_{h}\\cap[1, N]}f(x-n)\\Big|,\\ \\ \\mbox{for $x\\in\\mathbb{Z}$},$$ is of weak type $(1, 1)$, where $\\mathbf{N}_{h}=\\{n\\in\\mathbb{N}: \\exists_{m\\in\\mathbb{N}}\\ n=\\lfloor h(m)\\rfloor\\}$ for an appropriate function $h$. 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