{"paper":{"title":"A note on the set $\\boldsymbol{A(A+A)}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fran\\c{c}ois Hennecart, Ilya Shkredov, Pierre-Yves Bienvenu","submitted_at":"2018-11-21T18:28:14Z","abstract_excerpt":"Let $p$ a large enough prime number. When $A$ is a subset of $\\mathbb{F}_p\\smallsetminus\\{0\\}$ of cardinality $|A|> (p+1)/3$, then an application of Cauchy-Davenport Theorem gives $\\mathbb{F}_p\\smallsetminus\\{0\\}\\subset A(A+A)$. In this note, we improve on this and we show that if $|A|\\ge 0.3051 p$ implies $A(A+A)\\supseteq\\mathbb{F}_p\\smallsetminus\\{0\\}$. In the opposite direction we show that there exists a set $A$ such that $|A| > (1/8+o(1))p$ and $\\mathbb{F}_p\\smallsetminus\\{0\\}\\not\\subseteq A(A+A)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.08869","kind":"arxiv","version":2},"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"}