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Bourgain-type projection theorems over finite fields

Alex Rose

TL;DR

The paper develops finite-field analogs of Bourgain-type projection theorems in higher dimensions by introducing a non-degeneracy framework for families of subspaces and proving sharp projection bounds for one-dimensional and higher-codimension projections. It establishes a foundational line-projection case using incidence geometry, then leverages two robust reductions to extend to general codimensions, yielding explicit dependence on the size of the projection family $|E|$ and the ambient field. A final component handles very small sets in the plane via complex-analytic incindence techniques, completing a suite of Bourgain-type results over finite fields. These results advance the understanding of how fractal-like sets project under linear maps in finite-field geometries and sharpen prior bounds for exceptional projection configurations.

Abstract

We prove finite-field analogs of Bourgain's projection theorem in higher dimensions. In particular, for a certain range of parameters we improve on an exceptional set estimate by Chen in all dimensions and codimensions.

Bourgain-type projection theorems over finite fields

TL;DR

The paper develops finite-field analogs of Bourgain-type projection theorems in higher dimensions by introducing a non-degeneracy framework for families of subspaces and proving sharp projection bounds for one-dimensional and higher-codimension projections. It establishes a foundational line-projection case using incidence geometry, then leverages two robust reductions to extend to general codimensions, yielding explicit dependence on the size of the projection family and the ambient field. A final component handles very small sets in the plane via complex-analytic incindence techniques, completing a suite of Bourgain-type results over finite fields. These results advance the understanding of how fractal-like sets project under linear maps in finite-field geometries and sharpen prior bounds for exceptional projection configurations.

Abstract

We prove finite-field analogs of Bourgain's projection theorem in higher dimensions. In particular, for a certain range of parameters we improve on an exceptional set estimate by Chen in all dimensions and codimensions.

Paper Structure

This paper contains 9 sections, 23 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Over real numbers
  3. Over finite fields
  4. Main results
  5. Preliminaries and the outline of the proof
  6. Proof of the m = 1 case
  7. Proof of the reductions
  8. Proof of Theorem \ref{['BourgainTypeProjections']}
  9. Projections of very small sets and proof of Theorem \ref{['improvementTheorem']}

Key Result

Theorem 1.1

Let $K \subseteq \mathbb{R}^n$ be a Borel set. The following bounds hold:

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5: Corollary 1.3 in Che17
  • Conjecture 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 41 more