{"paper":{"title":"Online Beck--Fiala Down to Logarithmic Sparsity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","cs.DS","math.PR"],"primary_cat":"math.CO","authors_text":"Dylan J. Altschuler, Konstantin Tikhomirov","submitted_at":"2026-07-15T18:01:43Z","abstract_excerpt":"The Beck--Fiala conjecture asserts that every matrix $A\\in\\{0,1\\}^{n\\times T}$ with at most $d$ nonzero entries in each column has discrepancy $O(\\sqrt d)$. A major breakthrough result of Bansal and Jiang recently established the validity of the conjecture for $d \\ge \\log(T)^2$. The present article extends the validity of the classical \\textit{offline} Beck--Fiala conjecture to $d \\ge \\log(T)^{1+o(1)}$; moreover, the main thrust of the result is that it is actually obtained by an efficient \\textit{online} algorithm that minimizes prefix discrepancy. The result is also essentially optimal, sinc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14238","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/2607.14238/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"}