{"paper":{"title":"Online balancing of vectors with small coordinates","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.DM","cs.DS","math.PR"],"primary_cat":"math.CO","authors_text":"Antonios Hmadi","submitted_at":"2026-08-12T18:10:45Z","abstract_excerpt":"Let $v_1,\\ldots,v_T\\in B_2^m$ be fixed in advance and revealed sequentially, and assume that $\\|v_t\\|_\\infty\\leqslant d^{-1/2}$ for some $d\\geqslant 1$ and every $1\\leqslant t\\leqslant T$. There are absolute constants $L,C,c>0$ and a randomized online signing such that $$\\mathbb{P}\\left\\{\\max_{k\\leqslant T}\\left\\|\\sum_{t=1}^k\\varepsilon_t v_t\\right\\|_\\infty>6L\\right\\} \\leqslant CT\\exp\\left(-\\frac{cd}{\\ln^2(ed)}\\right).$$ Consequently, constant prefix discrepancy holds with probability at least $1-\\varepsilon$ once $d$ is at least $C\\ln\\frac{3T}{\\varepsilon}\\left[\\ln\\left(e+\\ln\\frac{3T}{\\vareps"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.12490","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/2608.12490/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"}