Estimates of character sums in finite fields
type: Item
instance of: scholarly article
title: Estimates of character sums in finite fields
zbMATH Open document ID: 1261.11085
DOI: 10.1134/S0001434610090221
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: Suppose \(p\) is a prime and \(\{\omega_1,\dots,\omega_n\}\) is a basis of \(\mathbb F_{p^n}\) over \(\mathbb F_p\). Suppose \(B\) is an \(n\)-dimensional parallelepiped with edges \(H_1,\dots,H_n\), that is \[ B=\left\{\sum_{x=1}^n x_i\omega_i:N_i+1\leq x_i\leq N_i+H_i, i=1,\dots,n\right\}, \] where \(0\leq N_i<N_i+H_i<p\). Suppose \(\chi\) is a nontrivial multiplicative character of \(\mathbb F_{p^n}\). The aim of the paper is to find non-trivial estimates of the character sum \(\sum_{x\in B} \chi(x)\). The main result is an extension of Burgess' classical estimate for character sums in the case \(n=1\). If \(0<\varepsilon\leq 1/4\) and \(p^{1/4+\varepsilon}\leq H_i\leq p\) for \(i=1,\dots,n\), then \[ |\sum_{x\in B} \chi(x)|\ll_{n,\varepsilon} p^{-\varepsilon^2/2}|B|. \] The first step in the proof uses a tour de force of the geometry of numbers to show that if \(H_1=\dots=H_n\leq p^{1/2}\), then the equation \(x^1x^2=x^3x^4\) with \(x^1, x^2, x^3, x^4\) in \(B\) has \(O_n(|B|^2\log p)\) solutions. The next step involves adding and subtracting parallelepipeds. For \(H\leq p/2\), let \(\Delta(H,\chi)=\max_B|\sum_{x\in B}\chi(x)|/|B|\) where the maximum is taken over parallelepipeds with \(H\leq H_i\leq 2H\) for \(i=1,\dots,n\). Since \(|\sum_{x\in B}\chi(x)|\leq \Delta(H,\chi)|B|\) and it is enough to bound \(\Delta([p^{1/2+\varepsilon}],\chi)\). If \(\tilde H\leq H/2\) and if \(x=\sum_{i=1}^n x_i\omega_i\) and \(u=\sum_{i=1}^n u_i\omega_i\) with \(1\leq u_i\leq \tilde H\) for \(i=1,\dots,n\), then \[ |\sum_{x\in B}\chi(x) - \sum_{x\in B}\chi(x+u)| \leq 6n\Delta(\tilde H,\chi)|B|\tilde H/H. \] A technique of Burgess then leads to an upper bound for \(\Delta(H,\chi)\) by means of the lemma from the first step and Schmidt's version of the Weil estimates for character sums. As a corollary, it follows that for all \(\varepsilon>0\) and \(\kappa>0\), there is a natural number \(k\) depending on \(\kappa\) and \(\varepsilon\) and a number \(c\) depending on \(n, \kappa\…
reviewed by: John H. Loxton
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11L15
zbMATH DE Number: 5980692
zbMATH Keywords: character sum
zbMATH Keywords: finite field
zbMATH Keywords: Burgess estimate
zbMATH Keywords: Weil estimate
zbMATH Keywords: Minkowski theorem
zbMATH Keywords: polar lattice
author: Sergei Konyagin
MaRDI profile type: Publication
cites work: On Character Sums and Primitive Roots†
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cites work: Q5572417
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full work available at URL: s0001434610090221
OpenAlex ID: W2061285505
description: scientific article; zbMATH DE number 5980692