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Parabolic distance in $\mathbb F_q^2$: a sharp exponent and new results

Dao Nguyen Van Anh, Steven Senger, Dung The Tran, Le Anh Vinh

Abstract

We study the parabolic variant of the Erd\H os--Falconer distance problem in finite fields. That is, if $q$ is odd, we seek size thresholds beyond which any subset $E\subset \mathbb F_q^2$ will determine many distinct parabolic distances. This problem has a rich history because the parabolic distance functional shares many properties with the standard distance functional, but exhibits many distinct behaviors. Here we begin with rather standard Fourier analytic arguments, but diverge into additive combinatorics to handle the central obstructions. We provide a suite of positive results and corresponding sharpness examples.

Parabolic distance in $\mathbb F_q^2$: a sharp exponent and new results

Abstract

We study the parabolic variant of the Erd\H os--Falconer distance problem in finite fields. That is, if is odd, we seek size thresholds beyond which any subset will determine many distinct parabolic distances. This problem has a rich history because the parabolic distance functional shares many properties with the standard distance functional, but exhibits many distinct behaviors. Here we begin with rather standard Fourier analytic arguments, but diverge into additive combinatorics to handle the central obstructions. We provide a suite of positive results and corresponding sharpness examples.
Paper Structure (9 sections, 15 theorems, 140 equations)

This paper contains 9 sections, 15 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The exponent of $\frac{3}{2}$
  4. Proof of Theorem \ref{['thm2']}
  5. Proof of Theorem \ref{['thm:main']}
  6. Some extensions of Theorem \ref{['thm:main']}
  7. A fiber-energy refinement
  8. A bipartite extension
  9. On the sharpness of Theorem \ref{['thm:main']}

Key Result

Theorem 1

The exponent $\frac{3}{2}$ is optimal. More precisely:

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Lemma 4
  • proof
  • Theorem 5
  • Theorem 6: Vinh LAV11
  • proof : Proof of Theorem \ref{['startingpoint']}
  • Proposition 7
  • proof
  • ...and 19 more