The Use in Additive Number Theory of Numbers without Large Prime Factors

In the past few years considerable progress has been made with regard to the known upper bounds for G(k) in Waring’s problem, that is, the smallest s such that every sufficiently large natural number is the sum of at most 8 kth powers of natural numbers. This has come about through the development of techniques using properties of numbers having only relatively small prime factors. In this article an account of these developments is given, and they are illustrated initially in a historical perspective through the special case of cubes. In particular the connection with the classical work of Davenport on smaller values of k is demonstrated. It is apparent that the fundamental ideas and the underlying mean value theorems and estimates for exponential sums have numerous applications and a brief account is given of some of them.