{"paper":{"title":"The structure of totally disconnected Host--Kra--Ziegler factors, and the inverse theorem for the $U^k$ Gowers uniformity norms on finite abelian groups of bounded torsion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.DS","authors_text":"Asgar Jamneshan, Or Shalom, Terence Tao","submitted_at":"2023-03-08T19:55:42Z","abstract_excerpt":"Let $\\Gamma$ be a countable abelian group, let $k\\geq 1$, and let $\\mathrm{X}=(X,\\mathcal{X},\\mu,T)$ be an ergodic $\\Gamma$-system of order $k$ in the sense of Host--Kra--Ziegler. The $\\Gamma$-system $\\mathrm{X}$ is said to be totally disconnected if all its structure groups are totally disconnected. We show that any totally disconnected $\\Gamma$-system of order $k$ is a generalized factor of a $\\mathbb{Z}^\\omega$-system with the structure of a Weyl system. As a consequence of this structure theorem, we show that totally disconnected $\\Gamma$-systems of order $k$ are represented by translation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.04860","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/2303.04860/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"}