On sets defining few ordinary circles

TL;DR

This paper investigates the minimum number of ordinary circles (circles passing through exactly three points) in large point sets in the plane, establishing exact bounds and characterizing extremal configurations, and extends to the circle variant of the orchard problem.

Contribution

It provides the first exact bounds and characterizations for the minimum and maximum number of ordinary and four-point circles in large point sets, linking these to algebraic curves.

Findings

Sets not on a line or circle have at least (1/4)n^2 - O(n) ordinary circles.

Maximum number of circles passing through exactly four points is (1/24)n^3 - O(n^2).

Most points in low-ordinary-circle sets lie on algebraic curves of degree at most four.

Abstract

An ordinary circle of a set $P$ of $n$ points in the plane is defined as a circle that contains exactly three points of $P$. We show that if $P$ is not contained in a line or a circle, then $P$ spans at least $\frac{1}{4}n^2 - O(n)$ ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large $n$ and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that $P$ spans at most $\frac{1}{24}n^3 - O(n^2)$ circles passing through exactly four points of $P$. Here we determine the exact maximum and the extremal configurations for all sufficiently large $n$. These results are based on the following structure theorem. If $n$ is sufficiently large depending on $K$, and $P$ is a set of $n$ points spanning at most $Kn^2$ ordinary circles, then all but $O(K)$ points of $P$ lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.