Explicit incidence bounds over general finite fields

2010/09/20 by Jones, Timothy G. F. · 1 citation
#11B75 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1009.3899

Abstract

Let \mathbbF_q be a finite field of order q=p^k where p is prime. Let P and L be sets of points and lines respectively in \mathbbF_q × \mathbbF_q with |P|=|L|=n. We establish the incidence bound I(P,L) ≤ γn^{3/2 - 1/12838}, where γ is an absolute constant, so long as P satisfies the conditions of being an `antifield'. We define this to mean that the projection of P onto some coordinate axis has no more than half-dimensional interaction with large subfields of \mathbbF_q. In addition, we give examples of sets satisfying these conditions in the important cases q=p² and q=p⁴.

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