New Estimates for Smooth Weyl Sums

Since the early part of this century, estimates for Weyl sums (or generalisations thereof) have been central to the treatment of many problems in the additive theory of numbers. For over forty years, the strongest such estimates have stemmed from a method due to Vinogradov [8], the argument having been somewhat simplified recently by the use of the large sieve (see [4, Lemma 5.4]). During this period, improvements in estimates for generalisations of Weyl sums have arisen from improved bounds on mean values of such sums, very recently with the arrival of Vaughan's new iterative method (see [5, Theorems 1.5 and 1.8]). In contrast, this paper will be devoted to improvements at the core of this circle of ideas, within Vinogradov's method itself. Our ideas, which here we shall investigate in the context of smooth Weyl sums, would seem to be applicable elsewhere, and this is a matter which we intend to pursue in the future. We now describe our conclusions in some detail. Let k be a natural number, and P be a large real number. When 2 < R < P, we define the set of /^-smooth numbers, s#(P, R), by