Areas of Triangles and SL2 Actions in Finite Rings

Abstract

In Euclidean space, one can use the dot product to give a formula for the area of
a triangle in terms of the coordinates of each vertex. Since this formula involves only addition,
subtraction, and multiplication, it can be used as a definition of area in R2
, where R is an
arbitrary ring. The result is a quantity associated with triples of points which is still invariant
under the action of SL2(R). One can then look at a configuration of points in R2
in terms of
the triangles determined by pairs of points and the origin, considering two such configurations
to be of the same type if corresponding pairs of points determine the same areas. In this paper
we consider the cases R = Fq and R = Z/p`Z , and prove that sufficiently large subsets of R2
must produce a positive proportion of all such types of configurations.