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Moment of double exponential sums

Nilanjan Bag, Dwaipayan Mazumder

TL;DR

This paper investigates the $2k$-th moment of double exponential sums in finite fields for the case $s=-1$ and $r=1$, focusing on $\mathcal{S}(\mathcal{M},\mathcal{N},k)=\sum_{m\in\mathcal{M}}\big|\sum_{n\in\mathcal{N}}{\bf e}_p(a\overline{m}n)\big|^{2k}$. The authors combine additive combinatorics (Fourier bias and additive energy) with an ab-shifting technique and Hölder inequalities to derive a nontrivial upper bound for general $\mathcal{M}$ (an interval) and $\mathcal{N}$ (an arbitrary subset):

Abstract

This paper is devoted to finding moments of double exponential sums with monomials over arbitrary sets and intervals in finite fields. The study of such sums dates back to the work of Heath-Brown, who studied such sums in a work on least square-free numbers in an arithmetic progression.

Moment of double exponential sums

TL;DR

This paper investigates the -th moment of double exponential sums in finite fields for the case and , focusing on . The authors combine additive combinatorics (Fourier bias and additive energy) with an ab-shifting technique and Hölder inequalities to derive a nontrivial upper bound for general (an interval) and (an arbitrary subset):

Abstract

This paper is devoted to finding moments of double exponential sums with monomials over arbitrary sets and intervals in finite fields. The study of such sums dates back to the work of Heath-Brown, who studied such sums in a work on least square-free numbers in an arithmetic progression.

Paper Structure

This paper contains 7 sections, 2 theorems, 40 equations.

Table of Contents

  1. Introduction and statements of the results
  2. Results from Additive combinatorics
  3. Fourier Bias
  4. Proof of Theorem \ref{['MT1']}
  5. Estimation of $\rho(l)$:
  6. Estimation of $\sum\limits_{n \in \mathfrak{N}}|\sum\limits_{m \in \mathcal{M}}{\bf{e}}_p(\overline{m}n)|$:
  7. Acknowledgement

Key Result

Theorem 1.1

Let $\mathcal{M}\subseteq\mathbb{F}_p^{\times}$ be an arbitrary interval of length $M$ and $\mathcal{N}\subset\mathbb{F}_p^{\times}$ be an arbitrary subset of cardinality $N$ with $\Sigma\neq \mathbb{F}_p$. Then for any fixed integer $r\geq 2$, we have which is non-trivial for $N> p^{\frac{1}{2}-\frac{1}{4rk}(1-\frac{1}{r})}.$

Theorems & Definitions (5)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: Uniformity implies large sum sets