{"paper":{"title":"Slow convergence of Birkhoff ergodic averages","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Valery V. Ryzhikov","submitted_at":"2025-07-22T16:25:07Z","abstract_excerpt":"Let $\\{T^z\\}$ be an ergodic action of the group $Z^n$ by automorphisms of the probability space $(X,m)$, $\\sum_{i}^\\infty a_i<\\infty$, $a_i>0$. For any sequence $M_k\\to +\\infty$ there exist $N_k>M_k$ and a function $ f\\in L_1(X,m)$ such that $$m\\left(\\ x:\\ \\left|\\, A(x,N_k,f) - \\int f \\, dm\\, \\right|\\ >\\ a_k \\ \\right)\\ \\to\\ 1,$$ where $$A(x,N,f) =\\frac 1 {N^n} \\sum_{z\\in Q_N} f(T^{z}x),$$ $$Q_N=\\{(z_1,\\dots,z_n)\\}\\, :\\, 1\\leq z_1,\\dots,z_n\\leq N\\}.$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.16740","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.16740/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}