{"paper":{"title":"Geometric Littlewood-Offord problems via lattice point counting","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT","math.PR"],"primary_cat":"math.CO","authors_text":"Alexandr Grebennikov, Matthew Kwan","submitted_at":"2025-05-30T15:22:29Z","abstract_excerpt":"Consider nonzero vectors $a_{1},\\dots,a_{n}\\in\\mathbb{C}^{k}$, independent Rademacher random variables $\\xi_{1},\\dots,\\xi_{n}$, and a set $S\\subseteq\\mathbb{C}^{k}$. What upper bounds can we prove on the probability that the random sum $\\xi_{1}a_{1}+\\dots+\\xi_{n}a_{n}$ lies in $S$? We develop a general framework that allows us to reduce problems of this type to counting lattice points in $S$. We apply this framework with known results from diophantine geometry to prove various bounds when $S$ is a set of points in convex position, an algebraic variety, or a semialgebraic set. In particular, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.24699","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/2505.24699/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"}