{"paper":{"title":"Almost Sure Bounds for Discrepancies of Linear Forms on the Circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hao Wu","submitted_at":"2021-11-29T21:51:24Z","abstract_excerpt":"As a generalization of irrational rotations and a dual case of higher-dimensional Kronecker sequences, we study the discrepancy of sequences of linear forms on the circle. Given irrationals $\\alpha_1,\\dots,\\alpha_d$, consider the set of $N^d$ points $\\{k_1\\alpha_1+\\cdots+k_d\\alpha_d \\mod 1 : 1\\le k_j\\le N\\}$. We prove that for a full-measure set of vectors $(\\alpha_1,\\dots,\\alpha_d)\\in\\mathbb{R}^d$, the maximal discrepancy of these points relative to intervals in $[0,1)$ has the optimal principal order $(\\log N)^d$, up to powers of $\\log\\log N$. This result provides a nearly sharp dual analogu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.14981","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/2111.14981/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"}