Quasi-diagonal Behaviour in Certain Mean Value Theorems of Additive Number Theory

Of fundamental significance in many problems of additive number theory are estimates for mean values of exponential sums over polynomial functions. In this paper we shall show that the exponential sums of greatest interest in additive number theory demonstrate quasi-diagonal behaviour, which is to say that by taking the degree of the polynomial argument sufficiently large, we can obtain upper bounds for low moments of the exponential sums arbitrarily close (in a suitable sense) to the diagonal estimate. Let us illustrate these notions by considering the situation in Vinogradov's mean value theorem. Let sand k be positive integers, and write