Burgess bounds for multi-dimensional short mixed character sums

Abstract

Abstract. This paper proves Burgess bounds for short mixed character sums in multidimensional settings. The mixed character sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial f , and a product of multiplicative Dirichlet characters. We combine a multi-dimensional Burgess method with recent results on multi-dimensional Vinogradov Mean Value Theorems for translation-dilation invariant systems in order to prove character sum bounds in k ≥ 1 dimensions that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding any given polynomial f into an advantageously chosen translation-dilation invariant system constructed in terms of f , we may in many cases significantly improve the bound for the associated character sum, due to a novel phenomenon that occurs only in dimensions k ≥ 2.

Burgess bounds for short character sums evaluated at forms II: the mixed case

“…In our present setting, we can see from the argument above that to exploit the homogeneity of F we must use the same prime p for each coordinate, leading to a smaller set to average over. Nevertheless, many of the arguments of [Pie16] may be adapted, and thus we will be efficient in our presentation.…”

“…Approximations of the additive character contribution. We first pass from T to T inside (3.4), by applying Lemma 3.3 of [Pie16] (an n-dimensional version of [BI86, Lemma 2]), which we recall here:…”

“…Earlier work on two special types of mixed sums in dimensions n ≥ 2 appeared in two recent papers. In the special case F (x) = x 1 • • • x n , the author proved nontrivial bounds for |S(F, g; N, H)| as long as H i > q 1/4+κ for some small κ > 0 [Pie16]. See also the preprint of Kerr [Ker14] in the case that F (x) = n i=1 L i (x) is the product of n linear forms L i that are linearly independent over F q .…”

“…Thus for any translation-dilation invariant system G for which a suitable bound is known for J r (G, X), we can deduce a Burgess bound for |S(F, g; N, H)|, for any polynomial g in the span of G. (Given g, one could optimize the choice of G as in [Pie16], but we do not pursue this here. )…”

Cancellations amongst Kloosterman sums

“…Similarly, for general quadratic polynomials f (X) = aX 2 + bX , with gcd(a, p) = 1, we can define the double sums Now, using the bound of Burgess [2] on short mixed sums (see [6,8,12] for various generalisations) we easily derive that for any fixed ν = 2, 3, . .…”