An explicit incidence theorem in F_p

2010/01/19 by Harald Andres Helfgott, Misha Rudnev · 1 citation
Mathematics · #math.CO #msc:11B75

paper · pdf · doi:10.1112/s0025579310001208

11 pages

arxiv created 2010/01/19 · arxiv updated 2014/01/14

Abstract

Let P = A× A ⊂ \mathbbF_p × \mathbbF_p, p a prime. Assume that P= A× A has n elements, n<p. See P as a set of points in the plane over \mathbbF_p. We show that the pairs of points in P determine ≥ c n^1 + 1/267 lines, where c is an absolute constant. We derive from this an incidence theorem: the number of incidences between a set of n points and a set of n lines in the projective plane over \F_p (n<√(p)) is bounded by C n^3/2-1/10678, where C is an absolute constant.

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