On triple intersections of three families of unit circles

TL;DR

This paper proves an improved upper bound on the number of points incident to circles from three families of unit circles passing through three fixed points, advancing understanding of geometric intersection problems.

Contribution

The paper establishes a tighter bound of O(n^{11/6}) for triple intersections of unit circles passing through three fixed points, improving previous results.

Findings

Number of triple intersection points is O(n^{11/6})

Improves previous bounds for this geometric problem

Addresses a conjecture by Székely

Abstract

Let $p_1,p_2,p_3$ be three distinct points in the plane, and, for $i=1,2,3$, let $\mathcal C_i$ be a family of $n$ unit circles that pass through $p_i$. We address a conjecture made by Sz\'ekely, and show that the number of points incident to a circle of each family is $O(n^{11/6})$, improving an earlier bound for this problem due to Elekes, Simonovits, and Szab\'o [Combin. Probab. Comput., 2009]. The problem is a special instance of a more general problem studied by Elekes and Szab\'o [Combinatorica, 2012] (and by Elekes and R\'onyai [J. Combin. Theory Ser. A, 2000]).