Incidence estimates for α-dimensional tubes and β-dimensional balls in R²

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+v/2≥ 1 , which is sharp when u+v ≥ 2 . We also show a new lower bound for the discretized sum-product problem.