A Szemerédi-Trotter type theorem, sum-product estimates in finite quasifields, and related results

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• 作者：Thang Pham^a ; ¹ ; ^{thang.pham@epfl.ch} ; Michael Tait^b ; ² ; ^{mtait@cmu.edu} ; Craig Timmons^c ; ³ ; ^{craig.timmons@csus.edu} ; Le Anh Vinh^d ; ⁴ ; ^{vinhla@vnu.edu.vn}
• 关键词：Szemer&eacute ; di&ndash ; Trotter theorem ; Quasifield ; Sum-product estimate
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：147
• 期：Complete
• 页码：55-74
• 全文大小：411 K
• 卷排序：147

文摘

We prove a Szemerédi–Trotter type theorem and a sum-product estimate in the setting of finite quasifields. These estimates generalize results of the fourth author, of Garaev, and of Vu. We generalize results of Gyarmati and Sárközy on the solvability of the equations a+b=cda+b=cd and ab+1=cdab+1=cd over a finite field. Other analogous results that are known to hold in finite fields are generalized to finite quasifields.