Incidence estimates for -dimensional tubes and -dimensional balls in R^2

We prove essentially sharp incidence estimates for a collection of -tubes and -balls in the plane, where the -tubes satisfy an -dimensional spacing condition and the -balls satisfy a -dimensional spacing condition. Our approach combines a combinatorial argument for small, and a Fourier analytic argument for large,. As an application, we prove a new lower bound for the size of a (u, v) -Furstenberg set when v 1, u+v/2 1, which is sharp when u+v 2. We also show a new lower bound for the discretized sum-product problem.