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It turns out that when $T$ is weakly mixing and $P_j(n,N)=p_jn+q_jN$ are linear or, more generally, have the form $P_j(n,N)=P_j(n)+Q_j(N)$ for some integer valued polynomials $P_j$ and $Q_j$ then the above averages converge in $L^2"},"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":"1702.05628","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-02-18T16:30:01Z","cross_cats_sorted":[],"title_canon_sha256":"80feae022080bacead3449e5448d1b62444094256f61c5640a1493fb2daf0673","abstract_canon_sha256":"2fdee644e095358beda56e48f0d32964a8df9c6ca65a6384548cc98750270282"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:18.898372Z","signature_b64":"x5kEV8zfitRrmEZoK8SVVWotTDsN73zF/UZE71HWTBz01Ro3yXxKArBc0BzN0QHwQatWQ9F0sEOlMqBRgR67Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cdb9f261d687a214cf2e3d14a65b4caa55c73f02381cedd20956933796abec4","last_reissued_at":"2026-05-18T00:29:18.897682Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:18.897682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ergodic Theorems for Nonconventional Arrays and an Extension of the Szemeredi Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Yuri Kifer","submitted_at":"2017-02-18T16:30:01Z","abstract_excerpt":"The paper is primarily concerned with the asymptotic behavior as $N\\to\\infty$ of averages of nonconventional arrays having the form $N^{-1}\\sum_{n=1}^N\\prod_{j=1}^\\ell T^{P_j(n,N)}f_j$ where $f_j$'s are bounded measurable functions, $T$ is an invertible measure preserving transformation and $P_j$'s are polynomials of $n$ and $N$ taking on integer values on integers. It turns out that when $T$ is weakly mixing and $P_j(n,N)=p_jn+q_jN$ are linear or, more generally, have the form $P_j(n,N)=P_j(n)+Q_j(N)$ for some integer valued polynomials $P_j$ and $Q_j$ then the above averages converge in $L^2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.05628","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":"1702.05628","created_at":"2026-05-18T00:29:18.897787+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.05628v6","created_at":"2026-05-18T00:29:18.897787+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.05628","created_at":"2026-05-18T00:29:18.897787+00:00"},{"alias_kind":"pith_short_12","alias_value":"TTNZ6JQ5NB5C","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_16","alias_value":"TTNZ6JQ5NB5CCTHS","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_8","alias_value":"TTNZ6JQ5","created_at":"2026-05-18T12:31:46.661854+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/TTNZ6JQ5NB5CCTHS4PIUUZNUZK","json":"https://pith.science/pith/TTNZ6JQ5NB5CCTHS4PIUUZNUZK.json","graph_json":"https://pith.science/api/pith-number/TTNZ6JQ5NB5CCTHS4PIUUZNUZK/graph.json","events_json":"https://pith.science/api/pith-number/TTNZ6JQ5NB5CCTHS4PIUUZNUZK/events.json","paper":"https://pith.science/paper/TTNZ6JQ5"},"agent_actions":{"view_html":"https://pith.science/pith/TTNZ6JQ5NB5CCTHS4PIUUZNUZK","download_json":"https://pith.science/pith/TTNZ6JQ5NB5CCTHS4PIUUZNUZK.json","view_paper":"https://pith.science/paper/TTNZ6JQ5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.05628&json=true","fetch_graph":"https://pith.science/api/pith-number/TTNZ6JQ5NB5CCTHS4PIUUZNUZK/graph.json","fetch_events":"https://pith.science/api/pith-number/TTNZ6JQ5NB5CCTHS4PIUUZNUZK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TTNZ6JQ5NB5CCTHS4PIUUZNUZK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TTNZ6JQ5NB5CCTHS4PIUUZNUZK/action/storage_attestation","attest_author":"https://pith.science/pith/TTNZ6JQ5NB5CCTHS4PIUUZNUZK/action/author_attestation","sign_citation":"https://pith.science/pith/TTNZ6JQ5NB5CCTHS4PIUUZNUZK/action/citation_signature","submit_replication":"https://pith.science/pith/TTNZ6JQ5NB5CCTHS4PIUUZNUZK/action/replication_record"}},"created_at":"2026-05-18T00:29:18.897787+00:00","updated_at":"2026-05-18T00:29:18.897787+00:00"}