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

TL;DR

This paper provides sharp estimates for the discretized Szemerédi-Trotter theorem and Furstenberg set problem in ^2, assuming certain spacing conditions on tubes, and constructs examples demonstrating these bounds.

Contribution

It introduces new sharp estimates for discretized geometric problems involving tubes in ^2 under spacing assumptions, along with explicit example constructions.

Findings

Sharp estimates for discretized Szemerédi-Trotter theorem

Sharp estimates for Furstenberg set problem

Construction of examples with shared features

Abstract

We study the $\delta$-discretized Szemer\'edi-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.