Some remarks on Gauss sums associated with kth powers

Abstract

Estimates for rational trigonometric sums are of great importance in analysing the local aspects of many additive problems. Indeed, bounds for the sumsin which e(α) denotes exp (2πiα), play a fundamental rôle in the application of the Hardy–Littlewood method to Waring's problem (see [11]), and also in the analysis of the local solubility of systems of additive equations (see, for example, [2]). When k ≥ 2 is an integer, and p is a prime number it is well known (see [5] or [11, lemma 4·3]) that