{"paper":{"title":"Sum-product estimates over arbitrary finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chun-Yen Shen, Doowon Koh, Sujin Lee, Thang Pham","submitted_at":"2018-05-23T00:03:42Z","abstract_excerpt":"In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $A\\subset \\mathbb{F}_q$ we have \\[|(A-A)^2+(A-A)^2|\\gg |A|^{1+\\frac{1}{21}}.\\] This can be viewed as the Erd\\H{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that \\[\\max\\{|A+A|, |A^2+A^2|\\}\\gg |A|^{1+\\frac{1}{42}}, ~|A+A^2|\\gg |A|^{1+\\frac{1}{84}}.\\]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.08910","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"}