On the number of ordinary circles

Abstract

We prove that any $n$ points in ${\mathbb{R}}^2$, not all on a line or circle, determine at least $\frac{1}{4}{n}^2-O(n)$ ordinary circles (circles containing exactly three of the $n$ points). The main term of this bound is best possible for even $n$. Our proof relies on a recent result of Green and Tao on ordinary lines.

v2: The previous version had a mistake that turned out to be fatal to the proof. The new version has a different proof of the same result, based on the same approach, but requiring a much more detailed analysis

Cited by in corpus (3)

• On sets defining few ordinary circles
• On sets defining few ordinary hyperplanes
• On the number of ordinary conics