{"paper":{"title":"The bad and rough rotation is Poissonian","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Manuel Hauke","submitted_at":"2025-06-02T14:46:01Z","abstract_excerpt":"Motivated by the Berry-Tabor Conjecture and the seminal work of Rudnick-Sarnak, the fine-scale properties of sequences $(a_n\\alpha)_{n \\in \\mathbb{N}} \\mod 1$ with $(a_n)_{n \\in \\mathbb{N}} \\subseteq \\mathbb{N} $ and $\\alpha$ irrational have been extensively studied in the last decades. In this article, we prove that for $(a_n)_{n \\in \\mathbb{N}}$ arising from the set of rough numbers with explicit roughness parameters and any badly approximable $\\alpha$, $(a_n\\alpha)_{n \\in \\mathbb{N}} \\mod 1$ has Poissonian correlations of all orders, and consequently, Poissonian gaps. This is the first know"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.01736","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/2506.01736/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"}