This proposal is concerned with bounds for mean values in additive number theory, and the local solubility of simultaneous additive equations. The principal investigator intends to improve estimates in Vinogradov's mean value theorem by incorporating the ideas behind the new refinements in the corresponding work in Waring's problem. These improvements will have consequences beyond additive number theory. This is research in the field of number theory. Number theory starts with the whole numbers and questions such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it has become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics.
