Tangency counting for well-spaced circles

Abstract

In the late 90's, Tom Wolff introduced the circle tangency counting problem in his expository article on the Kakeya conjecture. For collections of well-spaced circles, we break the $N^{3/2}$-barrier, proving that a set of $N$ well-spaced circles has at most $N^{25/18+\varepsilon}$ sites of internal tangency. The circle tangency problem can be related to a problem about incidences between points in $\mathbb{R}^3$ and light rays. For this problem, we introduce a stopping time argument to extract maximal information about well-spaced points from a refined decoupling theorem for the light cone in $\mathbb{R}^3$, leading to sharp bounds on the number of $\mu$-rich tangency rectangles.