{"paper":{"title":"A note on expansion in prime fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.CO","authors_text":"Laura Venieri, Tuomas Orponen","submitted_at":"2018-01-29T15:56:40Z","abstract_excerpt":"Let $\\beta,\\epsilon \\in (0,1]$, and $k \\geq \\exp(122 \\max\\{1/\\beta,1/\\epsilon\\})$. We prove that if $A,B$ are subsets of a prime field $\\mathbb{Z}_{p}$, and $|B| \\geq p^{\\beta}$, then there exists a sum of the form $$S = a_{1}B \\pm \\ldots \\pm a_{k}B, \\qquad a_{1},\\ldots,a_{k} \\in A,$$ with $|S| \\geq 2^{-12}p^{-\\epsilon}\\min\\{|A||B|,p\\}$.\n  As a corollary, we obtain an elementary proof of the following sum-product estimate. For every $\\alpha < 1$ and $\\beta,\\delta > 0$, there exists $\\epsilon > 0$ such that the following holds. If $A,B,E \\subset \\mathbb{Z}_{p}$ satisfy $|A| \\leq p^{\\alpha}$, $|"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09591","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":""},"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"}