Abstract

We prove essentially sharp incidence estimates for a collection of δ-tubes and δ-balls in the plane, where the δ-tubes satisfy an α-dimensional spacing condition and the δ-balls satisfy a β-dimensional spacing condition. Our approach combines a combinatorial argument for small α,β and a Fourier analytic argument for large α,β. As an application, we prove a new lower bound for the size of a (u,v)-Furstenberg set when v≥1,u+v2≥1, which is sharp when u+v≥2. We also show a new lower bound for the discretized sum-product problem.