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BRK-type sets over finite fields

Charlotte Trainor

Abstract

A Besicovitch-Rado-Kinney (BRK) set in $\mathbb{R}^n$ is a Borel set that contains a $(n-1)$-dimensional sphere of radius $r$, for each $r>0$. It is known that such sets have Hausdorff dimension $n$ from the work of Kolasa and Wolff. In this paper, we consider an analogous problem over a finite field, $\mathbb{F}_q$. We define BRK-type sets in $\mathbb{F}_q^n$, and establish lower bounds on the size of such sets using techniques introduced by Dvir's proof of the finite field Kakeya conjecture.

BRK-type sets over finite fields

Abstract

A Besicovitch-Rado-Kinney (BRK) set in is a Borel set that contains a -dimensional sphere of radius , for each . It is known that such sets have Hausdorff dimension from the work of Kolasa and Wolff. In this paper, we consider an analogous problem over a finite field, . We define BRK-type sets in , and establish lower bounds on the size of such sets using techniques introduced by Dvir's proof of the finite field Kakeya conjecture.
Paper Structure (15 sections, 15 theorems, 81 equations)

This paper contains 15 sections, 15 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Besicovitch-Rado-Kinney sets
  3. Analogous problem over finite fields
  4. Preliminaries
  5. Hasse Derivatives and the Method of Multiplicities
  6. Schwartz-Zippel Lemma and a key application
  7. Binomial coefficients and explicit form for Hasse derivatives
  8. The lexicographical order
  9. A key lemma
  10. Warm up argument in $\mathbb{F}_q^2$
  11. Proof of Theorem \ref{['thm']} assuming Proposition \ref{['prop']}
  12. Proof of Proposition \ref{['prop']}
  13. Proof in dimension $n=2$
  14. Proof in dimensions $n>2$
  15. Acknowledgements

Key Result

Theorem 1.2

Let $\ell\in \mathbb{N}$ with $2\leq \ell<q$. Let $S\subset\mathbb{F}_q^n$ be a BRK-type set of degree $\ell$, as defined in Definition def-brk. Then

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.1
  • Lemma 2.2: methodmult, Lemma 5
  • Lemma 2.3: methodmult, Proposition 6
  • Lemma 2.4: methodmult, Proposition 4
  • Lemma 2.5: methodmult, Lemma 8
  • Lemma 2.6: methodmult, Proposition 10
  • Lemma 2.7
  • ...and 14 more