On triple intersections of three families of unit circles

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 Székely, 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 Szabó [Combin. Probab. Comput., 2009]. The problem is a special instance of a more general problem studied by Elekes and Szabó [Combinatorica, 2012] (and by Elekes and Rónyai [J. Combin. Theory Ser. A, 2000]).

23 pages

References in corpus (3)

• Polynomials vanishing on Cartesian products: The Elekes-Szabó Theorem revisited
• Polynomials vanishing on grids: The Elekes-Rónyai problem revisited
• The number of unit-area triangles in the plane: Theme and variations

Cited by in corpus (1)

• Exposition of Elekes Szabo paper