{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:JCINZ3WNUS7RYXKHZUSCTPVCXN","short_pith_number":"pith:JCINZ3WN","schema_version":"1.0","canonical_sha256":"4890dceecda4bf1c5d47cd2429bea2bb55ff2784f7b7804a64099b572598d7c1","source":{"kind":"arxiv","id":"1612.05334","version":1},"attestation_state":"computed","paper":{"title":"Hochman's upcrossing theorem for groups of polynomial growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Nikita Moriakov","submitted_at":"2016-12-16T01:51:07Z","abstract_excerpt":"Consider a stochastic process $(S_{[a_i,b_i]})_{[a_i,b_i] \\subset \\mathbb{N}}$, which is indexed by the collection of all nonempty intervals $[a_i,b_i] \\subset \\mathbb{N}$ and which is stationary under translations of the intervals. It was shown by M. Hochman that, for any $k \\geq 1$ and any interval $(\\alpha,\\beta) \\subset \\mathbb{R}$, one can give an `almost-exponential' bound on the size of the set where the associated process $(S_{[1,n]})_{n \\geq 1}$ has at least $k$ fluctuations over $(\\alpha,\\beta)$. It was also noticed that a similar techniques can be applied in $\\mathbb{Z}^d$ case. In "},"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":"1612.05334","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-12-16T01:51:07Z","cross_cats_sorted":[],"title_canon_sha256":"cb0ef9dbe7038e2e1ffaeda7687e670d46581f6d125971c648a1eb5f13b6c59c","abstract_canon_sha256":"fd314a4319eae4bc1580ab92abcd6a98586c6b3a59386802b0edc17025d0ab0f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:52.041703Z","signature_b64":"qSACsmpA7/kDwPf6fVlgr7wHs4GIp2eD8GRDuK0gCqkkwkqCPl/jaZ5mwjPP+NBSgZbRcKzE/mxBtzbZ7S3SBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4890dceecda4bf1c5d47cd2429bea2bb55ff2784f7b7804a64099b572598d7c1","last_reissued_at":"2026-05-18T00:54:52.041310Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:52.041310Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hochman's upcrossing theorem for groups of polynomial growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Nikita Moriakov","submitted_at":"2016-12-16T01:51:07Z","abstract_excerpt":"Consider a stochastic process $(S_{[a_i,b_i]})_{[a_i,b_i] \\subset \\mathbb{N}}$, which is indexed by the collection of all nonempty intervals $[a_i,b_i] \\subset \\mathbb{N}$ and which is stationary under translations of the intervals. It was shown by M. Hochman that, for any $k \\geq 1$ and any interval $(\\alpha,\\beta) \\subset \\mathbb{R}$, one can give an `almost-exponential' bound on the size of the set where the associated process $(S_{[1,n]})_{n \\geq 1}$ has at least $k$ fluctuations over $(\\alpha,\\beta)$. It was also noticed that a similar techniques can be applied in $\\mathbb{Z}^d$ case. In "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.05334","kind":"arxiv","version":1},"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":"1612.05334","created_at":"2026-05-18T00:54:52.041377+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.05334v1","created_at":"2026-05-18T00:54:52.041377+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.05334","created_at":"2026-05-18T00:54:52.041377+00:00"},{"alias_kind":"pith_short_12","alias_value":"JCINZ3WNUS7R","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"JCINZ3WNUS7RYXKH","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"JCINZ3WN","created_at":"2026-05-18T12:30:25.849896+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/JCINZ3WNUS7RYXKHZUSCTPVCXN","json":"https://pith.science/pith/JCINZ3WNUS7RYXKHZUSCTPVCXN.json","graph_json":"https://pith.science/api/pith-number/JCINZ3WNUS7RYXKHZUSCTPVCXN/graph.json","events_json":"https://pith.science/api/pith-number/JCINZ3WNUS7RYXKHZUSCTPVCXN/events.json","paper":"https://pith.science/paper/JCINZ3WN"},"agent_actions":{"view_html":"https://pith.science/pith/JCINZ3WNUS7RYXKHZUSCTPVCXN","download_json":"https://pith.science/pith/JCINZ3WNUS7RYXKHZUSCTPVCXN.json","view_paper":"https://pith.science/paper/JCINZ3WN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.05334&json=true","fetch_graph":"https://pith.science/api/pith-number/JCINZ3WNUS7RYXKHZUSCTPVCXN/graph.json","fetch_events":"https://pith.science/api/pith-number/JCINZ3WNUS7RYXKHZUSCTPVCXN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JCINZ3WNUS7RYXKHZUSCTPVCXN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JCINZ3WNUS7RYXKHZUSCTPVCXN/action/storage_attestation","attest_author":"https://pith.science/pith/JCINZ3WNUS7RYXKHZUSCTPVCXN/action/author_attestation","sign_citation":"https://pith.science/pith/JCINZ3WNUS7RYXKHZUSCTPVCXN/action/citation_signature","submit_replication":"https://pith.science/pith/JCINZ3WNUS7RYXKHZUSCTPVCXN/action/replication_record"}},"created_at":"2026-05-18T00:54:52.041377+00:00","updated_at":"2026-05-18T00:54:52.041377+00:00"}