{"paper":{"title":"Conditional expanding bounds for two-variables functions over prime fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fran\\c{c}ois Hennecart, Norbert Hegyv\\'ari","submitted_at":"2013-09-29T12:10:01Z","abstract_excerpt":"In this paper we provide in $\\bFp$ expanding lower bounds for two variables functions $f(x,y)$ in connection with the product set or the sumset. The sum-product problem has been hugely studied in the recent past. A typical result in $\\bFp^*$ is the existenceness of $\\Delta(\\alpha)>0$ such that if $|A|\\asymp p^{\\alpha}$ then $$ \\max(|A+A|,|A\\cdot A|)\\gg |A|^{1+\\Delta(\\alpha)}, $$ Our aim is to obtain analogous results for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$: if $|A|\\asymp|B|\\asymp p^{\\alpha}$ then $$ \\max(|f(A,B)|,|g(A,B)|)\\gg |A|^{1+\\Delta(\\alpha)} $$ for some $\\Delta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7580","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"}