{"paper":{"title":"Pointwise multiple averages for systems with two commuting transformations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Sebastian Donoso, Wenbo Sun","submitted_at":"2015-09-30T19:42:52Z","abstract_excerpt":"We show that if $(X,\\mathcal{X},\\mu,S,T)$ is an ergodic measure preserving system with commuting transformations $S$ and $T$, then the average \\[\\frac{1}{N^3} \\sum_{i,j,k=0}^{N-1} f_0(S^j T^k x) f_1 (S^{i+j} T^k x) f_2 (S^j T^{i+k} x)\\] converges for $\\mu$-a.e. $x\\in X$ as $N\\to \\infty$ for $f_0,f_1, f_2\\in L^\\infty(\\mu)$. We also show that if $(X,\\mathcal{X},\\mu,S,T)$ is a measurable distal system, the average \\[ \\frac{1}{N}\\sum_{i=0}^{N-1} f_1 (S^i x) f_2 (T^i x) \\] converges for $\\mu$-a.e. $x\\in X$ as $N\\to \\infty$ for $f_1,f_2\\in L^{\\infty}(\\mu)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.09310","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"}