Estimates of character sums in finite fields
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 \(n, \kappa\) and \(\varepsilon\) such that if \(A\subset B\) and \(|A|\geq\kappa|B|\), then \(|A^k|\geq cp^n\).
scientific article; zbMATH DE number 5980692