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arXiv · 2511.21656

Simple proofs of discretised projection theorems

William O'Regan·Pablo Shmerkin·Hong Wang
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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.

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William O'Regan, Pablo Shmerkin, Hong Wang. 2025-11-26. Simple proofs of discretised projection theorems. https://arxiv.org/abs/2511.21656

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