On Exponential Sums and Additive Diophantine Equations

这一建议涉及加性数论中值的界，以及联立加性方程的局部可解性。主要研究人员打算通过将维诺格拉多夫中值定理背后的思想融入到瓦林问题的相应工作中来改进中值定理中的估计。这些改进将产生超越加法数论的后果。这是数论领域的研究。数论始于整数和诸如一个整数被另一个整数整除的问题。它是最古老的数学领域之一，最初是出于纯粹的美学原因。然而，在过去的半个世纪里，它已经成为开发计算机科学的新算法和电子产品的新纠错码的必要工具。

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.