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

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 $v \ge 1, u + \frac{v}{2} \ge 1$, which is sharp when $u + v \ge 2$. We also show a new lower bound for the discretized sum-product problem.