{"paper":{"title":"On growth rate in $SL_2(\\mathbf{F}_p)$, the affine group and sum-product type implications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ilya D. Shkredov, Misha Rudnev","submitted_at":"2018-12-04T20:36:43Z","abstract_excerpt":"This paper aims to study in more depth the relation between growth in matrix groups ${{\\rm SL_2}}(\\mathbf{F})$ and ${{\\rm Aff}}(\\mathbf{F})$ over a field $\\mathbf{F}$ by multiplication and geometric incidence estimates, associated with the sum-product phenomenon over $\\mathbf{F}$. It presents streamlined proofs of Helfgott's theorems on growth in the $\\mathbf{F}_p$-case, which avoid sum-product estimates. For ${{\\rm SL_2}}(\\mathbf{F}_p)$, for sets exceeding in size some absolute constant, we improve the lower bound $\\frac{1}{1512}$ for the growth exponent, due to Kowalski, to $\\frac{1}{21}.$ F"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.01671","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"}