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We show that for \"typical\" choices of Hardy field functions $a(t)$ with polynomial growth, the averages $\\frac{1}{N}\\sum_{n=1}^N f_1(T^{[a(n)]}x)\\cdot...\\cdot f_\\ell(T^{\\ell [a(n)]}x)$ converge in the mean and we determine their limit. For example, this is the case if $a(t)=t^{3/2}, t\\log{t},$ or $t^2+(\\log{t})^2$. Furthermore, if ${a_1(t),...,a_\\ell(t)}$ is a \"typical\" family of logarithmico-exponential functions of polynomial growth, "},"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":"0903.0042","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-02-28T13:48:16Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0b8d1d76a97ebe980717741f1dceb0c9575ef70b132d52a313a4592075095f38","abstract_canon_sha256":"0e34383f3bdd8c6f92ca4df098defa3f2cc44e88b6beffe6c034af055846649d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:03.064334Z","signature_b64":"i0CXEKmwCWSBdA1ZE9XguGJJ0K+fZMKX7FhPhcwxeZCXe5nTM6JRBq+hDwxEi5YGvmTbhMZvI7YCIWeyKYuDAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29b959c9805abea09a90b4dae968209624b0ee4513097c2585df1346a7aebedc","last_reissued_at":"2026-05-18T03:38:03.063827Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:03.063827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple recurrence and convergence for Hardy sequences of polynomial growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.DS","authors_text":"Nikos Frantzikinakis","submitted_at":"2009-02-28T13:48:16Z","abstract_excerpt":"We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for \"typical\" choices of Hardy field functions $a(t)$ with polynomial growth, the averages $\\frac{1}{N}\\sum_{n=1}^N f_1(T^{[a(n)]}x)\\cdot...\\cdot f_\\ell(T^{\\ell [a(n)]}x)$ converge in the mean and we determine their limit. For example, this is the case if $a(t)=t^{3/2}, t\\log{t},$ or $t^2+(\\log{t})^2$. Furthermore, if ${a_1(t),...,a_\\ell(t)}$ is a \"typical\" family of logarithmico-exponential functions of polynomial growth, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.0042","kind":"arxiv","version":6},"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":"0903.0042","created_at":"2026-05-18T03:38:03.063906+00:00"},{"alias_kind":"arxiv_version","alias_value":"0903.0042v6","created_at":"2026-05-18T03:38:03.063906+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.0042","created_at":"2026-05-18T03:38:03.063906+00:00"},{"alias_kind":"pith_short_12","alias_value":"FG4VTSMALK7K","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_16","alias_value":"FG4VTSMALK7KBGUQ","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_8","alias_value":"FG4VTSMA","created_at":"2026-05-18T12:25:59.703012+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/FG4VTSMALK7KBGUQWTNOS2BASY","json":"https://pith.science/pith/FG4VTSMALK7KBGUQWTNOS2BASY.json","graph_json":"https://pith.science/api/pith-number/FG4VTSMALK7KBGUQWTNOS2BASY/graph.json","events_json":"https://pith.science/api/pith-number/FG4VTSMALK7KBGUQWTNOS2BASY/events.json","paper":"https://pith.science/paper/FG4VTSMA"},"agent_actions":{"view_html":"https://pith.science/pith/FG4VTSMALK7KBGUQWTNOS2BASY","download_json":"https://pith.science/pith/FG4VTSMALK7KBGUQWTNOS2BASY.json","view_paper":"https://pith.science/paper/FG4VTSMA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0903.0042&json=true","fetch_graph":"https://pith.science/api/pith-number/FG4VTSMALK7KBGUQWTNOS2BASY/graph.json","fetch_events":"https://pith.science/api/pith-number/FG4VTSMALK7KBGUQWTNOS2BASY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FG4VTSMALK7KBGUQWTNOS2BASY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FG4VTSMALK7KBGUQWTNOS2BASY/action/storage_attestation","attest_author":"https://pith.science/pith/FG4VTSMALK7KBGUQWTNOS2BASY/action/author_attestation","sign_citation":"https://pith.science/pith/FG4VTSMALK7KBGUQWTNOS2BASY/action/citation_signature","submit_replication":"https://pith.science/pith/FG4VTSMALK7KBGUQWTNOS2BASY/action/replication_record"}},"created_at":"2026-05-18T03:38:03.063906+00:00","updated_at":"2026-05-18T03:38:03.063906+00:00"}