Incidence estimates for $\alpha$-dimensional tubes and $\beta$-dimensional balls in ${\mathbb{R}}^2$

Abstract

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