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Simple proofs of discretised projection theorems

William O'Regan, Pablo Shmerkin, Hong Wang

TL;DR

The paper provides a streamlined, self-contained proof of Bourgain’s discretised projection theorem, emphasizing a direct expansion mechanism under weak two-ends spacing rather than structure-theorem prerequisites. Central to the approach is a short, elementary expansion result built from a lattice-translation/slab method and Marstrand-type projections, which then yields ring-type growth and a discretised projection bound via a sequence of additive-combinatorial reductions and projections. The results consolidate and simplify key tools in geometric measure theory, harmonic analysis, and homogeneous dynamics, offering a more accessible entry point to discretised projection theory and its applications to problems such as Furstenberg sets and Kakeya-type questions. The methods combine an Edgar–Miller-inspired viewpoint with a concise use of projection theorems (Marstrand, Kaufman) and Balog–Szemerédi–Gowers, producing a compact chain from two-ends spacing to global projection estimates with wide applicability.

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.

Simple proofs of discretised projection theorems

TL;DR

The paper provides a streamlined, self-contained proof of Bourgain’s discretised projection theorem, emphasizing a direct expansion mechanism under weak two-ends spacing rather than structure-theorem prerequisites. Central to the approach is a short, elementary expansion result built from a lattice-translation/slab method and Marstrand-type projections, which then yields ring-type growth and a discretised projection bound via a sequence of additive-combinatorial reductions and projections. The results consolidate and simplify key tools in geometric measure theory, harmonic analysis, and homogeneous dynamics, offering a more accessible entry point to discretised projection theory and its applications to problems such as Furstenberg sets and Kakeya-type questions. The methods combine an Edgar–Miller-inspired viewpoint with a concise use of projection theorems (Marstrand, Kaufman) and Balog–Szemerédi–Gowers, producing a compact chain from two-ends spacing to global projection estimates with wide applicability.

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 satisfying a weak ``two-ends'' spacing condition is expanded by a polynomial to a set of positive Lebesgue measure.

Paper Structure

This paper contains 10 sections, 17 theorems, 100 equations, 1 figure.

Table of Contents

  1. Introduction and statement of main results
  2. Introduction
  3. Main results
  4. Preliminaries
  5. Notation
  6. Additive combinatorics
  7. Geometric measure theory
  8. Proof of Theorem \ref{['thm.expansion']}
  9. Proof of Theorem \ref{['thm.ring']}
  10. Proof of Theorem \ref{['thm.projhaus']}

Key Result

Theorem 1.1

Let $0 < \kappa < 1$ and let $C >0$. Then there exists an integer $N$ depending on $C$ and $\kappa$ only so that the following holds. Let $\mu$ be a $(\kappa,C)$-measure supported on $[-C,C]$. Write $K := \operatorname{spt} \mu$. Then,

Figures (1)

  • Figure 1: Slab $T$ of width and radius $\sim 1$ containing mutually separated translates of $A^n$, and such that $\pi(T)$ is bounded. Once the number of translates is large enough (but still $\sim 1$), the projection $\pi$ can no longer be injective on their union. This allows us to eliminate one of the $v_j$ from the sumset, at the cost of replacing $A$ by $N A^{(2)}- N A^{(2)}$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: Triangle inequality
  • Proposition 2.2: Plünnecke--Ruzsa inequality
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 2.5: Balog-Szemerédi-Gowers
  • ...and 19 more