{"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"}