On point-line incidences in vector spaces over finite fields
Let Fq F q be the finite field of q q elements. We show that for almost every point set P P and line set L L in F q 2 of cardinality |P|=|L|?q | P | = | L | ? q , there exists a pair (p,l)?PL ( p , l ) ? P  L with p?l p ? l . We also obtain a similar result in the setting of the finite cyclic ring Z/mZ Z / m Z .
point-line incidences szemerditrotter theorem residue rings