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The natural interpretation of even moments of exponential sums, in terms of the number of solutions of certain underlying diophantine equations, permits a rich interplay to be developed between simple analytic inequalities, and estimates for those even moments. This interplay is in large part responsible for the remarkable success enjoyed by the Hardy-Littlewood method in its application to numerous problems of additive type. In the absence of such an interpretation, the most effective method for bounding odd and fractional moments, hitherto, has been to apply H61der's inequality to interpolate linearly between the exponents arising at even moments. The object of this paper is to establish a method for handling all moments of exponential sums over smooth numbers non-trivially, thereby breaking out of the latter (classically) implied convex region of permissible exponents. In view of the great flexibility and applicability of the new iterative methods of Vaughan and Wooley (see, for example, [13, 16, 18]), this breakthrough has many consequences. In this paper we confine ourselves to two relatively accessible applications, deriving new bounds for sums of cubes, and strengthening substantially what is known about quasi-diagonal behaviour.

Breaking classical convexity in Waring's problem: Sums of cubes and quasi-diagonal behaviour