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Bourgain's projection theorem over normed division algebras

William O'Regan

Abstract

We give a simple and self-contained proof of an extension of a projection theorem of Bourgain over the reals to division algebras over local fields of zero characteristic.

Bourgain's projection theorem over normed division algebras

Abstract

We give a simple and self-contained proof of an extension of a projection theorem of Bourgain over the reals to division algebras over local fields of zero characteristic.

Paper Structure

This paper contains 15 sections, 11 theorems, 192 equations.

Table of Contents

  1. Introduction and statement of results
  2. Introduction
  3. Statement of results
  4. Preliminaries
  5. Basic definitions
  6. Uniform subsets
  7. Escape from subspaces
  8. Expansion under addition and multiplication
  9. Parameters
  10. Non-field case
  11. Field case
  12. An iterative expansion estimate
  13. Projections of Cartesian products
  14. Projections of general sets
  15. Proof of Theorem \ref{['thm.adultproj2']}

Key Result

Theorem 1.1

Let $d$ be a positive integer. Let $E$ be a normed division algebra of dimension $d$ over $\mathbb{Q}_p$ or $\mathbb{R}.$ Let $0 < s \leq \sigma < d, t > 0.$ There exists $c = c(s,\sigma,d,t)> 0$ so that the following holds for all $\delta,\epsilon >0$ small enough. Let $A \subset B(0,1)$ be a $(\d Further, provided that $E$ is not a non-commutative division algebra over $\mathbb{Q}_p,$ then $c =

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['prop.uniform']}
  • Theorem 2.2
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm.expand']}
  • Theorem 3.2
  • ...and 12 more