On a question of Davenport and Lewis and new character sum bounds in finite fields
In this paper the author obtains new character sum bounds in finite fields \(\mathbb F_q\) with \(q=p^n\), using methods from additive combinatorics related to the sum-product phenomenon. More precisely, Burgess' classical amplification argument is combined with his estimate on the `multiplicative energy' for subsets in \(\mathbb F_q\). The latter appears as a quantitative version of the sum-product theorem in finite fields (see [\textit{J. Bourgain, N. Katz} and \textit{T. Tao}, Geom. Funct. Anal. 14, No. 1, 27--57 (2004; Zbl 1145.11306)] and [\textit{T. Tao} and \textit{V. H. Vu}, Additive Combinatorics, Cambridge University Press (2006; Zbl 1127.11002)]) following arguments from [\textit{M. Garaev}, Int. Math. Res. Not. 2007, No. 11, Article ID rnm035, 11 p. (2007; Zbl 1160.11014)], [\textit{N. H. Katz} and \textit{C.-Y. Shen}, Proc. Am. Math. Soc. 136, No. 7, 2499--2504 (2008; Zbl 1220.11013)] and [Online J. Anal. Comb. 3, Article 3, 6 p. (2008; Zbl 1241.11023)]. The first results relate to the work [Rend. Circ. Mat. Palermo (2) 12, 129-136 (1963; Zbl 0119.04302)] of \textit{H. Davenport} and \textit{D. J. Lewis}. Let \(\chi\) be a nontrivial multiplicative character of \(\mathbb F_{p^n}\). Then the following results are proved. (1) Let \(\varepsilon >0\) be given. If \[ B=\left\{\sum_{j=1}^n x_j\omega_j: x_j\in [N_j+1,N_j+H_j]\cap\mathbb Z,\, j=1,\dots,n\right\} \] is a box satisfying \(\prod_{j=1}^n H_j> p^{(\frac 25 +\varepsilon)n}\), then for \(p>p(\varepsilon)\) and some absolute constant \(c>0\), he proves, denoting \(\chi\) a nontrivial multiplicative character \[ \left| \sum_{x\in B}\chi(x)\right| < cnp^{-\frac{\varepsilon^2}4}| B| \] unless \(n\) is even, \(\chi\) is principal on a subfield \(F_2\) of size \(p^{n/2}\) and \(\max_\xi | B\cap\xi F_2|>p^{-\varepsilon}| B|\). (2) Assume \(A,B\subset\mathbb F_p\) such that \(| A|>p^{\frac 49 +\varepsilon}\), \(| B|>p^{\frac 49 +\varepsilon}\), \(| B+B|< k\,| B|\). Then \[ \left|\sum_{x\in A, y\in B} \chi(x+y)\right|<p^{-\tau}| A|\,| B|. \] The novelty of the method in this paper is to exploit the finite field combinatorics without the need to reduce the problem to a divisor issue in \(\mathbb Z\) or in the integers of an algebraic number field \(K\). (3) Let \(I\subset\mathbb F_p\) be an interval with \(| I|=p^\beta\) and let \(\mathcal D\subset\mathbb F_p\) be a \(p^\beta\)-spaced set with \(| \mathcal D| =p^\sigma\). Assume \(\beta>\tfrac14-\tfrac{\sigma}{4(1-\sigma)}+\delta\). Then for a non-principal multiplicative character \(\chi\) he proves \[ \left|\sum_{x\in I, y\in \mathcal D} \chi(x+y)\right| < p^{-\frac{\delta^2}{4}}| I|\,| \mathcal D|. \] He also improves a result of \textit{A. A. Karatsuba} [Sov. Math., Dokl. 44, No. 1, 145--148 (1992); translation from Dokl. Akad. Nauk SSSR 319, No. 3, 543--545 (1991; Zbl 0772.11028)].
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