A new form of the circle method, and its application to quadratic forms.

Abstract

If the coefficients r(n) satisfy suitable arithmetic conditions the behaviour of F (α) will be determined by an appropriate rational approximation a/q to α, with small values of q usually producing large values of F (α). When α lies in an interval [a/q − δ, a/q + δ] with q small, a ‘major arc’, one hopes to estimate F (α) asymptotically, while if the corresponding q is large, for the ‘minor arcs’, one hopes that F (α) will be small, at least on average. One can distinguish two different types of application of the circle method. The first type, following the above pattern, uses one of a variety of mean-value techniques, Hua’s inequality for example, to give an upper bound for the minor arc contribution. Results on Waring’s problem are generally proved this way, for example. The second type of application, on which we shall be focusing our attention, avoids the minor arcs completely, and treats F (α) asymptotically for all α. Here one cannot take advantage of the mean-value ideas used in the first type of problem. The savings come instead from the availability of good estimates for ‘complete’ exponential sums. Kloosterman’s work [12] on quaternary quadratic forms, and the papers of the author [5] and Hooley [8] and [9] on cubic forms, are of this latter type. We should point out that such applications do not include any problems in which F is the generating function for a polynomial of degree 4 or more. There are