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Exceptional projections in finite fields: Fourier analytic bounds and incidence geometry

Jonathan M. Fraser, Firdavs Rakhmonov

TL;DR

The paper develops a Fourier-analytic framework for finite-field projection problems, establishing an $L^p$-based bound on the number of exceptional projections and thereby strengthening finite-field Marstrand-type results. It introduces $(p,s)$-Salem sets, proves a subspace Plancherel theorem and a new subspace character-sum identity (extending Chen to general finite fields), and applies these tools to incidences between point sets and affine $k$-planes, yielding a self-contained incidence bound without spectral graph theory. The work also provides sharpness examples across incidence regimes and demonstrates a novel route from incidence estimates back to projection bounds, including a uniform lower-bound result for projections. Collectively, these results advance finite-field harmonic analysis, incidence geometry, and the transfer of geometric-measure-theoretic ideas to discrete settings, with implications for related combinatorial problems.

Abstract

We consider the problem of bounding the number of exceptional projections (projections which are smaller than typical) of a subset of a vector space over a finite field onto subspaces. We establish bounds that depend on $L^p$ estimates for the Fourier transform, improving various known bounds for sets with sufficiently good Fourier analytic properties. The special case $p=2$ recovers a recent result of Bright and Gan (following Chen), which established the finite field analogue of Peres--Schlag's bounds from the continuous setting. We prove several auxiliary results of independent interest, including a character sum identity for subspaces (solving a problem of Chen) and a full generalization of Plancherel's theorem for subspaces. These auxiliary results also have applications in affine incidence geometry, that is, the problem of estimating the number of incidences between a set of points and a set of affine $k$-planes. We present a novel and direct proof of a well-known result in this area that avoids the use of spectral graph theory, and we provide simple examples demonstrating that these estimates are sharp up to constants.

Exceptional projections in finite fields: Fourier analytic bounds and incidence geometry

TL;DR

The paper develops a Fourier-analytic framework for finite-field projection problems, establishing an -based bound on the number of exceptional projections and thereby strengthening finite-field Marstrand-type results. It introduces -Salem sets, proves a subspace Plancherel theorem and a new subspace character-sum identity (extending Chen to general finite fields), and applies these tools to incidences between point sets and affine -planes, yielding a self-contained incidence bound without spectral graph theory. The work also provides sharpness examples across incidence regimes and demonstrates a novel route from incidence estimates back to projection bounds, including a uniform lower-bound result for projections. Collectively, these results advance finite-field harmonic analysis, incidence geometry, and the transfer of geometric-measure-theoretic ideas to discrete settings, with implications for related combinatorial problems.

Abstract

We consider the problem of bounding the number of exceptional projections (projections which are smaller than typical) of a subset of a vector space over a finite field onto subspaces. We establish bounds that depend on estimates for the Fourier transform, improving various known bounds for sets with sufficiently good Fourier analytic properties. The special case recovers a recent result of Bright and Gan (following Chen), which established the finite field analogue of Peres--Schlag's bounds from the continuous setting. We prove several auxiliary results of independent interest, including a character sum identity for subspaces (solving a problem of Chen) and a full generalization of Plancherel's theorem for subspaces. These auxiliary results also have applications in affine incidence geometry, that is, the problem of estimating the number of incidences between a set of points and a set of affine -planes. We present a novel and direct proof of a well-known result in this area that avoids the use of spectral graph theory, and we provide simple examples demonstrating that these estimates are sharp up to constants.

Paper Structure

This paper contains 20 sections, 13 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Basics of discrete Fourier analysis
  4. Gaussian binomial coefficients
  5. Main results: exceptional projections in finite fields
  6. Character sums for subspaces and a problem of Chen
  7. Plancherel's theorem for subspaces
  8. Gaussian binomial coefficients: proof of Lemma \ref{['relation between G(k,n) and GBC']}
  9. Exceptional projections: proof of Theorem \ref{['main projection theorem']}
  10. Uniform lower bounds: proof of Theorem \ref{['main projection theorem2']}
  11. Another application: incidences in Fqn via Fourier analysis
  12. Background and context
  13. Main results on incidences
  14. Sharpness examples
  15. Examples with few incidences
  16. ...and 5 more sections

Key Result

Lemma 2.3

Let $k,n\in \mathbb{N}_0$. Then, the following statements hold:

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Corollary 3.5
  • proof : Proof of Corollary \ref{['main cor']}
  • Theorem 3.6
  • ...and 19 more