Geometric incidence theorems via Fourier analysis

TL;DR

This paper establishes a connection between bounds on generalized Radon transforms and discrete incidence theorems for manifolds and regular point sets using Fourier analysis.

Contribution

It introduces a novel approach linking Fourier analysis of Radon transforms to incidence geometry, providing new bounds and insights.

Findings

Non-trivial bounds for Radon transforms imply incidence theorems

Fourier analysis techniques are effective in incidence geometry

Results apply to manifolds and regular point sets

Abstract

We prove that non-trivial bounds for generalized Radon transforms imply correspondingly non-trivial discrete incidence theorems for manifolds and suitably regular point sets.