{"paper":{"title":"Expanding phenomena over matrix rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Chun-Yen Shen, Doowon Koh, Le Anh Vinh, Thang Pham, Ye\\c{s}\\.im Dem\\.iro\\u{g}lu Karabulut","submitted_at":"2018-03-22T13:48:59Z","abstract_excerpt":"In this paper, we study expanding phenomena in the setting of matrix rings. More precisely, we will prove that\n  If $A$ is a set of $M_2(\\mathbb{F}_q)$ and $|A|\\gg q^{7/2}$, then we have \\[|A(A+A)|, ~|A+AA|\\gg q^4.\\]\n  If $A$ is a set of $SL_2(\\mathbb{F}_q)$ and $|A|\\gg q^{5/2}$, then we have \\[|A(A+A)|, ~|A+AA|\\gg q^4.\\]\n  We also obtain similar results for the cases of $A(B+C)$ and $A+BC$, where $A, B, C$ are sets in $M_2(\\mathbb{F}_q)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.08357","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":""},"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"}