Areas of triangles and Beck's theorem in planes over finite fields (Q313454)

This paper establishes that any subset \(E \subseteq {\mathbb{F}}_q^2\) of a plane over the finite field \({\mathbb{F}}_q\) with \(| E| > q\) determines at least \(\frac{q-1}{2}\) distinct areas of triangles. Moreover, such triangles exist sharing a common base in \(E\), and hence a common vertex. A finite field version of Beck's theorem is established, which shows that under the more stringent condition \(| E| = \Omega(q \log q)\), there are at least \(q-o(q)\) distinct areas of triangles sharing a common vertex \(z\), for a positive proportion of \(z\in E\). Higher-dimensional implications of these results are also discussed.