{"paper":{"title":"New results on sum-product type growth over fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Brendan Murphy, Giorgis Petridis, Ilya D. Shkredov, Misha Rudnev, Oliver Roche-Newton","submitted_at":"2017-02-03T13:39:01Z","abstract_excerpt":"We prove a range of new sum-product type growth estimates over a general field $\\mathbb{F}$, in particular the special case $\\mathbb{F}=\\mathbb{F}_p$. They are unified by the theme of \"breaking the $3/2$ threshold\", epitomising the previous state of the art. These estimates stem from specially suited applications of incidence bounds over $\\mathbb{F}$, which apply to higher moments of representation functions.\n  We establish the estimate $|R[A]| \\gtrsim |A|^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios defined by triples of elements of a (sufficiently small if $\\mathbb{F}$ h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.01003","kind":"arxiv","version":5},"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"}