On Exponential Sums and Additive Diophantine Equations

这一建议涉及的是 加法数论，和局部溶解度的同时 加法方程 首席研究员打算改善 在维诺格拉多夫中值定理中， 在相应的工作中新的改进背后的想法， 华林的问题。 这些改进将产生的影响超出 加法数论 这是数论领域的研究。 Number 理论开始于整数和问题，如 一个整数被另一个整数整除。 它是其中的 最古老的数学领域，它最初是追求 纯粹的美学原因。 然而，在过去的半个世纪里， 它已经成为开发新算法的重要工具， 计算机科学和新的电子纠错码。

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.