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Distribution of solutions to systems of congruences in balls

Michael Harm

Abstract

Let $G_1,\dots, G_n\in \mathbb{F}_p[X_1,\dots,X_m]$ be $n$ polynomials in $m$ variables over the finite field $\mathbb{F}_p$ of $p$ elements. For any sufficiently large prime $p$ and non-trivial bounds for the Weyl sums associated to the non-trivial linear combinations of $G=(G_1,\dots, G_n)$, we study various properties regarding the distribution of the vectors by fractional parts \begin{equation*} \bigg(\bigg\{ \frac{G_1(\textbf{x})}{p}\bigg\},\cdots,\bigg\{ \frac{G_n(\textbf{x})}{p}\bigg\}\bigg)\in \mathbb{T}^n,\hspace{10pt} \textbf{x}\in \mathbb{F}_p^m. \end{equation*} We prove refinements of equidistribution, such as bounds for the ball discrepancy and variance.

Distribution of solutions to systems of congruences in balls

Abstract

Let be polynomials in variables over the finite field of elements. For any sufficiently large prime and non-trivial bounds for the Weyl sums associated to the non-trivial linear combinations of , we study various properties regarding the distribution of the vectors by fractional parts \begin{equation*} \bigg(\bigg\{ \frac{G_1(\textbf{x})}{p}\bigg\},\cdots,\bigg\{ \frac{G_n(\textbf{x})}{p}\bigg\}\bigg)\in \mathbb{T}^n,\hspace{10pt} \textbf{x}\in \mathbb{F}_p^m. \end{equation*} We prove refinements of equidistribution, such as bounds for the ball discrepancy and variance.
Paper Structure (12 sections, 11 theorems, 60 equations)

This paper contains 12 sections, 11 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Equidistribution
  3. Discrepancy
  4. Variance
  5. Preliminaries
  6. Fourier transform
  7. A Bessel function of the first order
  8. Indicator function of Lg
  9. A Zeta function
  10. Proofs
  11. Proof of Theorem \ref{['thm: shrinking target']}
  12. Proof of Theorem \ref{['thm: variance']}

Key Result

Theorem 1.5

Let $G$ be a polynomial system of type $\eta\geq 0$ (Definition def: eta). Furthermore let $B_R(y)$ be a Euclidean $n$-ball with radius $0< R<1/2$ and center point $y\in {\mathbb T}^n$ (if given). We have where $\mu$ denotes the Lebesgue measure.

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Theorem 1.5
  • Definition 1.6
  • Corollary 1.7
  • Definition 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 20 more