Point line incidences over finite fields

Sophie Stevens (Bristol)

Frank Adams 1,

Given finite point and line sets defined in the real (or complex) plane, we can bound the maximum number of times that these sets can intersect -- the Szemerédi-Trotter theorem. This bound has been applied to get bounds on seemingly-unrelated areas, such as sum-product theory. Can we do the same for finite point and lines sets defined over any finite field to say something non-trivial? Assuming that the point and line sets are not too big, previous literature answers this in the affirmative by using tools from additive combinatorics. I will present a new, stronger bound which circumvents many of these tools, returning to the geometry of the question. I will then give some of the applications of this incidence bound. This is joint work with Frank de Zeeuw.
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