{"paper":{"title":"The non-ergodic Host-Kra-Ziegler structure theorem for $\\mathbb{Z}^d$-actions via measurable selections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Asgar Jamneshan, Simon Machado","submitted_at":"2026-01-14T15:13:51Z","abstract_excerpt":"We establish a non-ergodic version of the Host-Kra-Ziegler structure theorem for measure-preserving $\\mathbb{Z}^d$-actions. Our argument reduces the non-ergodic case to the ergodic theorem (for $d\\ge 2$ due to Candela and Szegedy) via a measurable selection procedure. We also establish a non-ergodic vertical nilcharacter version of our main result. The non-ergodic version of the Host-Kra-Ziegler structure theorem is a key input in the companion paper by the second author and Fraczyk classifying point processes (i.e. random subsets) of $\\mathbb{Z}^d$ whose law is invariant under the group $\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.09553","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.09553/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"}