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For $d,k\\in \\mathbb{N}$, let $p_{i,j}(n), 1\\le i\\le k, 1\\le j\\le d$ be polynomials with rational coefficients taking integer values on the integers and $p_{i,j}(0)=0$. We show that if the expressions $g_i(n)=T_1^{p_{i,1}(n)}\\cdots T_d^{p_{i,d}(n)}$ depends nontrivially on $n$ for $i=1,2,\\cdots,k$, and for all $i\\neq j\\in \\{1,2,\\ldots,k\\}$ the expressions $g_i(n)g_j(n)^{-1}$ depend nontrivially on $"},"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":"1604.07113","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-04-25T02:42:45Z","cross_cats_sorted":[],"title_canon_sha256":"c4fdb92162cd7842936b6d0d6c7ba2ff8b0e1846bb088cf05d20013e6f88fde4","abstract_canon_sha256":"f8cd8687195e3d861d6ffc380326ca385fabbad1cf2badde73d032441792cebd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:58.506747Z","signature_b64":"4afZgYC5Too8q3glfa6cyb6dB9PrImhbJ1SrDUcKs0B0rX6XLyzfZhIR6JOfmER8SqsiNgma8goRhyes/A0/Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4441cafc0fbab5e66ee8e15d030560b710743210889150cd143e927f647621ec","last_reissued_at":"2026-05-18T00:59:58.506250Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:58.506250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topological correspondence of multiple ergodic averages of nilpotent group actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Song Shao, Wen Huang, Xiangdong Ye","submitted_at":"2016-04-25T02:42:45Z","abstract_excerpt":"Let $(X,\\Gamma)$ be a topological system, where $\\Gamma$ is a nilpotent group generated by $T_1,\\ldots, T_d$ such that for each $T\\in \\Gamma$, $T\\neq e_\\Gamma$, $(X,T)$ is weakly mixing and minimal. For $d,k\\in \\mathbb{N}$, let $p_{i,j}(n), 1\\le i\\le k, 1\\le j\\le d$ be polynomials with rational coefficients taking integer values on the integers and $p_{i,j}(0)=0$. We show that if the expressions $g_i(n)=T_1^{p_{i,1}(n)}\\cdots T_d^{p_{i,d}(n)}$ depends nontrivially on $n$ for $i=1,2,\\cdots,k$, and for all $i\\neq j\\in \\{1,2,\\ldots,k\\}$ the expressions $g_i(n)g_j(n)^{-1}$ depend nontrivially on $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07113","kind":"arxiv","version":2},"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":"1604.07113","created_at":"2026-05-18T00:59:58.506334+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.07113v2","created_at":"2026-05-18T00:59:58.506334+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.07113","created_at":"2026-05-18T00:59:58.506334+00:00"},{"alias_kind":"pith_short_12","alias_value":"IRA4V7APXK26","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"IRA4V7APXK26M3XI","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"IRA4V7AP","created_at":"2026-05-18T12:30:22.444734+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/IRA4V7APXK26M3XI4FOQGBLAW4","json":"https://pith.science/pith/IRA4V7APXK26M3XI4FOQGBLAW4.json","graph_json":"https://pith.science/api/pith-number/IRA4V7APXK26M3XI4FOQGBLAW4/graph.json","events_json":"https://pith.science/api/pith-number/IRA4V7APXK26M3XI4FOQGBLAW4/events.json","paper":"https://pith.science/paper/IRA4V7AP"},"agent_actions":{"view_html":"https://pith.science/pith/IRA4V7APXK26M3XI4FOQGBLAW4","download_json":"https://pith.science/pith/IRA4V7APXK26M3XI4FOQGBLAW4.json","view_paper":"https://pith.science/paper/IRA4V7AP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.07113&json=true","fetch_graph":"https://pith.science/api/pith-number/IRA4V7APXK26M3XI4FOQGBLAW4/graph.json","fetch_events":"https://pith.science/api/pith-number/IRA4V7APXK26M3XI4FOQGBLAW4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IRA4V7APXK26M3XI4FOQGBLAW4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IRA4V7APXK26M3XI4FOQGBLAW4/action/storage_attestation","attest_author":"https://pith.science/pith/IRA4V7APXK26M3XI4FOQGBLAW4/action/author_attestation","sign_citation":"https://pith.science/pith/IRA4V7APXK26M3XI4FOQGBLAW4/action/citation_signature","submit_replication":"https://pith.science/pith/IRA4V7APXK26M3XI4FOQGBLAW4/action/replication_record"}},"created_at":"2026-05-18T00:59:58.506334+00:00","updated_at":"2026-05-18T00:59:58.506334+00:00"}