Tangency counting for well-spaced circles

In this paper, we study a discretized variant of the circle tangency counting problem introduced by Tom Wolff in his expository article on the Kakeya conjecture. For collections of well-spaced circles, we introduce a stopping time argument to extract maximal information from a refined decoupling theorem for the cone in $\mathbb{R}^3$, leading to sharp bounds on the number of $\mu$-rich tangency rectangles. As a consequence, we break the $N^{3/2}$-barrier for exact circle tangencies when the circles are well-spaced.