{"paper":{"title":"Tight lower bounds for anti-concentration of Rademacher sums and Tomaszewski's counterpart problem","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Julien Portier, Lawrence Hollom","submitted_at":"2023-06-13T14:39:09Z","abstract_excerpt":"In this paper we prove that $\\mathbb{P}(|X| \\geq \\sqrt{\\text{Var}(X)}) \\geq 7/32$ for every finite Rademacher sum $X$, confirming a conjecture by Hitczenko and Kwapie{\\'n} from 1994, and improving upon results from Burkholder, Oleszkiewicz, and Dvo\\v{r}\\'ak and Klein. Moreover we fully determine the function $f(y)= \\inf_X \\mathbb{P}(|X| \\geq y\\sqrt{\\text{Var}(X)})$ where the $\\inf$ is taken over all finite Rademacher sums $X$, confirming a conjecture by Lowther and giving a partial answer to a question by Keller and Klein."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.07811","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/2306.07811/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"}