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

The first main result of this paper establishes that any sufficiently large subset of a plane over the finite field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{F}_q$$\end{document}, namely any set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subseteq \mathbb{F}_q^2$$\end{document} of cardinality |E| > q, determines at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{{q - 1}} {2}$$\end{document} 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.