{"paper":{"title":"A Weyl-type theorem for Diophantine approximations driven by LCA groups and applications","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"Every action of a locally compact Abelian group on the torus decomposes into uniquely ergodic subsystems.","cross_cats":["math.CA","math.NT"],"primary_cat":"math.DS","authors_text":"Aihua Fan","submitted_at":"2026-05-15T03:44:30Z","abstract_excerpt":"We investigate actions of locally compact Abelian (LCA) groups on the torus $\\mathbb{T}^n$, motivated by their close connection with Diophantine approximation. While Kronecker's theorem yields a classical density result, we prove a stronger equidistribution theorem of Weyl type: every such action admits a decomposition into uniquely ergodic subsystems. The proof of this result is based on a characterization of unique ergodicity for actions of amenable groups on compact metric spaces. As consequences, we establish several foundational results for LCA groups, including the Bohr orthogonality of "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Every action of a locally compact Abelian group on the torus admits a decomposition into uniquely ergodic subsystems, yielding a Weyl-type equidistribution theorem.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The proof relies on a pre-existing characterization of unique ergodicity for amenable group actions on compact metric spaces; if this characterization fails to apply to the specific LCA actions considered here, the decomposition result does not follow.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Establishes a decomposition of LCA group actions on the torus into uniquely ergodic subsystems, with applications to Bohr orthogonality and Wiener-type theorems on LCA groups.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every action of a locally compact Abelian group on the torus decomposes into uniquely ergodic subsystems.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0ecc38cbd9738dce05012dd030d042cd6719495293304d868410ed5bd6865385"},"source":{"id":"2605.15580","kind":"arxiv","version":1},"verdict":{"id":"74e245a9-ec46-4e2a-9c14-a10b481102aa","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T19:55:00.275916Z","strongest_claim":"Every action of a locally compact Abelian group on the torus admits a decomposition into uniquely ergodic subsystems, yielding a Weyl-type equidistribution theorem.","one_line_summary":"Establishes a decomposition of LCA group actions on the torus into uniquely ergodic subsystems, with applications to Bohr orthogonality and Wiener-type theorems on LCA groups.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The proof relies on a pre-existing characterization of unique ergodicity for amenable group actions on compact metric spaces; if this characterization fails to apply to the specific LCA actions considered here, the decomposition result does not follow.","pith_extraction_headline":"Every action of a locally compact Abelian group on the torus decomposes into uniquely ergodic subsystems."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15580/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T20:01:42.984127Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T20:01:19.289739Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T19:34:35.248320Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:41:56.070628Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"a6583efc4456fcc74ae169ac6590813bfe349dc5db44e0f6341d9d1426f6b8c7"},"references":{"count":29,"sample":[{"doi":"","year":1971,"title":"L. 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