Extremal secant size distributions in projective planes

Abstract

Given a point set $S$ in a projective plane of order $q$, each line $\ell$ determines a secant of size $|S\cap \ell|$. We study how concentrated this secant-size distribution can be. We develop a general framework for estimating and, in several cases, determining the extremal number. We point out how the extremal constructions vary depending on the remainder of the size of the set modulo $q$, and classify several extremal families of point sets, building from concurrent lines, Baer {subplanes} and lines in general position. On the lower bound side, we rely on stability results in the Desarguesian case and applications of the standard equalities in general. Finally, we connect the problem to a novel concept, called general position on level 2 for point and line sets, which then leads to a Sidon-type problem over finite fields called projective Sidon sets, providing an additional arithmetic model for highly concentrated secant-size distributions.