{"paper":{"title":"Growth in linear groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Brendan Murphy, Endre Szab\\'o, L\\'aszl\\'o Pyber, Sean Eberhard","submitted_at":"2021-07-14T13:05:19Z","abstract_excerpt":"We prove a conjecture of Helfgott on the structure of sets of bounded tripling in bounded rank, which states the following. Let $A$ be a finite symmetric subset of $\\mathrm{GL}_n(\\mathbf{F})$ for any field $\\mathbf{F}$ such that $|A^3| \\leq K|A|$. Then there are subgroups $H \\trianglelefteq \\Gamma \\trianglelefteq \\langle A \\rangle$ such that $A$ is covered by $K^{O_n(1)}$ cosets of $\\Gamma$, $\\Gamma/H$ is nilpotent of step at most $n-1$, and $H$ is contained in $A^{O_n(1)}$. This theorem includes the Product Theorem for finite simple groups of bounded rank as a special case. As an application "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.06674","kind":"arxiv","version":3},"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/2107.06674/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"}