Economical Waring basesVan H. VuUniversity of California, San Diego
March 20,

ABSTRACT


Few hundreds years ago, Waring asserted (without proof) that every atural number can be represented as the sum of few powers. More precisely, for every natural number k, there is a nutural number s such that every natural number n can be written as sum of s kth power (of nonnegative integers). For instance, every natural number is a sum of 4 squares, 9 ubes and so on. Waring's assertion has turn into one of the main research problems in number theory. It was first proved by Hilbert, and a little bit later by Hardy and Littlewood, using the circle method. Works on Waring's problem still continue even today. About 20 years ago, Nathanson posed a question that whether one can represent all natural numbers using only "few" powrs (by few we mean a sparse subset of the set of all kth powers). May results have been obtained by Erdos, Nathanson, Zollner and Wirsing for the case k=2. In this talk, we will solve Nathanson's question for general k. The proof is a combination of number theory, probablity and combinatorics and some of the tools arof independent interest. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

