Areas of triangles and SL_2 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 $R^2$, where $R$ is an arbitrary ring. The result is a quantity associated with triples of points which is still invariant under the action of $\text{SL}_2(R)$. One can then look at a configuration of points in $R^2$ 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 $R=\mathbb{F}_q$ and $R=\mathbb{Z}/p^\ell \mathbb{Z}$, and prove that sufficiently large subsets of $R^2$ must produce a positive proportion of all such types of configurations.