Simple proofs of discretised projection theorems

TL;DR

This paper presents a simplified, self-contained proof of Bourgain's discretised projection theorem, introducing an elementary argument that demonstrates how certain discretised sets expand under polynomial transformations, impacting geometric measure theory and harmonic analysis.

Contribution

It provides a new, elementary proof of a key discretised projection theorem, making the result more accessible and easier to apply in related fields.

Findings

Discretised sets satisfying a two-ends spacing condition expand under polynomial maps.

The proof simplifies understanding of Bourgain's theorem and its applications.

The approach enhances tools available in geometric measure theory and harmonic analysis.

Abstract

We give a simple, short and self-contained presentation of Bourgain's discretised projection theorem from 2010, which is a fundamental tool in many recent breakthroughs in geometric measure theory, harmonic analysis, and homogeneous dynamics. Our main innovation is a short elementary argument that shows that a discretised subset of $\R$ satisfying a weak ``two-ends'' spacing condition is expanded by a polynomial to a set of positive Lebesgue measure.