Incidence estimates for quasi-product sets and applications

TL;DR

This paper leverages recent advances in Furstenberg set theory to establish new incidence bounds for discretized Cartesian product structures, impacting fractal measures, energy estimates, and sum-product problems.

Contribution

It introduces novel incidence results for discretized structures with Cartesian product features, advancing understanding of fractal measures and sum-product phenomena.

Findings

New incidence bounds of Szemerédi–Trotter type for discretized Cartesian products

Progress on energy estimates and Fourier decay for fractal measures on curves

Enhanced sum-product estimates governed by fractal dimension

Abstract

We use recent advances in the theory of Furstenberg sets to prove new incidence results of Szemer\'edi--Trotter strength for $\delta$-discretized structures with Cartesian product flavor. We use these results to make progress on a number of problems that include energy estimates and Fourier decay of fractal measures supported on curves, as well as various sum-product-like results governed by fractal dimension.