{"paper":{"title":"Slowly decaying averages and fat towers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"James T. Campbell, M\\'at\\'e Wierdl","submitted_at":"2016-09-18T22:10:44Z","abstract_excerpt":"Let $(X,\\Sigma,m,\\tau)$ be an ergodic system, that is, $(X, \\Sigma, m)$ is a probability space and $\\tau: X \\to X$ is an invertible ergodic $m$-preserving transformation. For a function $f:X\\to\\mathbb R$, let $A_Nf$ denote the $N$th ergodic average, $A_Nf(x)=\\frac{1}{N}\\cdot (f(x)+\\dots+\\tau^ {N-1}f(x))$. Martin Barlow (personal communication) asked the following question, which arose from the work of a student (Zichun Ye) on interface models. Question: If $f(x) \\ge 0$ is integrable, and $N(x) = \\min \\{n: A_kf(x) \\le 2 \\int f \\text{for all} k \\ge n\\}$, is it the case that $N(x)$ is also integr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05560","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"}