{"paper":{"title":"Algebraic aspects of the polynomial Littlewood-Offord problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT","math.PR"],"primary_cat":"math.CO","authors_text":"Lisa Sauermann, Matthew Kwan, Yiting Wang, Zhihan Jin","submitted_at":"2025-05-29T10:53:39Z","abstract_excerpt":"Consider a degree-$d$ polynomial $f(\\xi_1,\\dots,\\xi_n)$ of independent Rademacher random variables $\\xi_1,\\dots,\\xi_n$. To what extent can $f(\\xi_1,\\dots,\\xi_n)$ concentrate on a single point? This is the so-called polynomial Littlewood-Offord problem. A nearly optimal bound was proved by Meka, Nguyen and Vu: the point probabilities are always at most about $1/\\sqrt n$, unless $f$ is \"close to the zero polynomial\" (having only $o(n^d)$ nonzero coefficients).\n  In this paper we prove several results supporting the general philosophy that the Meka-Nguyen-Vu bound can be significantly improved un"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.23335","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/2505.23335/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"}