{"paper":{"title":"Bourgain's $L^2$ pointwise ergodic theorem over function fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Andrew Lott, Th\\'ai Ho\\`ang L\\^e","submitted_at":"2026-05-27T18:53:57Z","abstract_excerpt":"We prove a function-field analogue of Bourgain's $L^2$ pointwise ergodic theorem. Let $q$ be a power of a prime $p$, let $\\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\\mathbb{F}_q$, and let $\\mathbb{F}_q[t][u]$ be the ring of polynomials over $\\mathbb{F}_q[t]$. Let $T^{(1)},\\ldots,T^{(\\ell)}$ be commuting, measure-preserving $\\mathbb{F}_q[t]$-actions on a $\\sigma$-finite measure space $(X,\\mu)$, and let $P_1,\\ldots,P_\\ell\\in \\mathbb{F}_q[t][u]\\setminus\\{0\\}$. Define a sequence of operators $(A_n)_{n\\in \\mathbb{N}}$ by \\[ A_n g(x):=\\frac{1}{q^n}\\sum_{\\substack{f\\in \\mathbb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28997","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/2605.28997/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"}