{"paper":{"title":"Maximal Gaps for Dilated Lacunary Integer Sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.PR"],"primary_cat":"math.NT","authors_text":"Bohan Yang, Yuval Peres","submitted_at":"2026-06-27T10:54:00Z","abstract_excerpt":"Let \\((a_n)_{n\\ge1}\\subset\\mathbb{N}\\) be a lacunary sequence, \\(a_{n+1}\\ge q a_n\\) for \\(q>1\\). For \\(x\\in\\mathbb{T}\\), we study the maximal empty circular gap \\(G_N(x)\\) of the finite orbit \\(\\{a_1x,\\ldots,a_Nx\\}\\). We prove that, for Lebesgue-almost every \\(x\\), \\[\n  \\frac{1}{2}\n  \\le \\liminf_{N\\to\\infty}\\frac{NG_N(x)}{\\log N}\n  \\le \\limsup_{N\\to\\infty}\\frac{NG_N(x)}{\\log N}\n  \\le \\frac{q+1}{q-1}\\,. \\] If, in addition, \\(a_n\\mid a_{n+1}\\) for every \\(n\\), then this can be improved to \\[\n  \\lim_{N\\to\\infty}\\frac{NG_N(x)}{\\log N}=1 \\] for Lebesgue-almost every \\(x\\)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28860","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/2606.28860/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"}