Point sets determining few angles are almost contained in a line or circle

Abstract

We prove a structural theorem for point sets in ${\mathbb{R}}^2$ which determine few pinned angles. More precisely, we prove the existence of an absolute constant $c>0$ such that if $n$ is sufficiently large and $P$ is a set of $n$ points then there exists a point $q\in P$ which determines at least $n^{1+c}$ distinct angles to other pairs of points of $P$, provided that $P$ is not of one of the following exceptional forms: all but at most one of the points of $P$ lie on a line; all but two points of $P$ lie on a line, and the two exceptional points are symmetric with respect to the line; all the points of $P$ lie on a circle; all but one of the points of $P$ lie on a circle, and the exceptional point is the centre of the circle. As a consequence, we answer a question of Corrádi, Erdős and Hajnal by showing that if $n$ is sufficiently large and $P\subseteq {\mathbb{R}}^2$ has cardinality $n$ and is not contained on a single line, then $P$ determines at least $n-2$ angles. Moreover, we prove that the unique point set attaining this minimum is the regular $n$-gon.