29 - Recent Work on Waring's Problem

Abstract

This chapter focuses on the recent work on Waring's problem. The central problem in Waring's problem is the determination of G(k), that is, the smallest number s such that every large natural number is a sum of at most s positive k-th powers. The chapter discusses a theorem of technical nature, which indicates that the underlying techniques might have wider applicability. The bound given by the theorem might be compared with that obtaining from Vinogradov's mean value theorem for the classical generating function in Waring's problem. The technique puts no obstacle in the way of methods that have been developed in the context of diminishing ranges.