Szemerédi-Trotter type results in arbitrary finite fields

2018/08/16 by Mohammadi, Ali
#11B75 #52C35 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.05543

Abstract

Let q be a power of a prime and \mathbbF_q the finite field consisting of q elements. We prove explicit upper bounds on the number of incidences between lines and Cartesian products in \mathbbF_q². We also use our results on point-line incidences to give new sum-product type estimates concerning sums of reciprocals.

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• Further improvements to incidence and Beck-type bounds over prime finite fields
• Multi-parameter Szemerédi-Trotter-type theorems and applications in finite fields
• The Szemerédi-Trotter theorem over arbitrary field of characteristic zero
• Szemeredi-Trotter type theorem and sum-product estimate in finite fields
• Explicit incidence bounds over general finite fields