An incidence estimate and a Furstenberg type estimate for tubes in ${\mathbb{R}}^2$

Abstract

We study the $\delta$-discretized Szemerédi-Trotter theorem and Furstenberg set problem. We prove sharp estimates for both two problems assuming tubes satisfy some spacing condition. For both two problems, we construct sharp examples that have many common features.

23 pages; to appear in Journal of Fourier Analysis and Applications