On point-line incidences in vector spaces over finite fields

Let FqFq be the finite field of qq elements. We show that for almost every point set PP and line set LL in Fq2 of cardinality |P|=|L|≳q|P|=|L|≳q, there exists a pair (p,l)∈P×L(p,l)∈P×L with p∈lp∈l. We also obtain a similar result in the setting of the finite cyclic ring Z/mZZ/mZ.