Areas of triangles and Beck’s theorem in planes over finite fields

Abstract

The first main result of this paper establishes that any sufficiently large subset of a plane over the finite field $\mathbb{F}_q$, namely any set $E \subseteq \mathbb{F}_q^2$ of cardinality |E| > q, determines at least $\tfrac{{q - 1}} {2}$ distinct areas of triangles. Moreover, one can find such triangles sharing a common base in E, and hence a common vertex. However, we stop short of being able to tell how “typical” an element of E such a vertex may be.

It is also shown that, under a more stringent condition |E| = Ω(q log q), there are at least q − o(q) distinct areas of triangles sharing a common vertex z, this property shared by a positive proportion of z ∈ E. This comes as an application of the second main result of the paper, which is a finite field version of the Beck theorem for large subsets of $\mathbb{F}_q^2$. Namely, if |E| = Ω(q log q), then a positive proportion of points z ∈ E has a property that there are Ω(q) straight lines incident to z, each supporting, up to constant factors, approximately the expected number $\tfrac{{\left| E \right|}} {q}$ of points of E, other than z. This is proved by combining combinatorial and Fourier analytic techniques. A counterexample in [14] shows that this cannot be true for every z ∈ E; unless $\left| E \right| = \Omega \left( {q^{\tfrac{3} {2}} } \right)$.

We also briefly discuss higher-dimensional implications of these results in light of some recent developments in the literature.