A stability theorem for lines in Galois planes of prime order

In this paper we prove that a point set of size less than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{3}{2}(q+1)}$$\end{document} in PG(2, q), q prime, that has relatively few 0-secants must contain many collinear points. More precise bounds can be found in Theorem 4.