Nonstandard Analysis in Additive Number Theory - An Unconventional Approach to Upper Density or Upper Banach Density Problems

主要研究者：本研究课题是关于非标准分析在可加数论中的应用，特别是在上密度或上Banach密度问题中的应用。在加法数论中有许多关于Shnirel'man密度或下密度的有趣定理，在组合数论中有一些关于上密度或上Banach密度的有趣定理。然而，在可加数论中处理上密度或上Banach密度仍然是一个未知领域。在这一领域的主要未触及的问题之一是寻找正上密度或正上Banach密度的集合和的增长性和结构。主要研究者一直在使用非标准分析研究这个问题，并获得了许多有趣的新结果。建议的第一部分描述了一个拓扑的方法来解决这个问题。利用超有限Loeb空间上的U-拓扑，主要研究者证明了非标准分析中的一个强大定理，这意味着一些标准数学领域中的几个新定理，包括加法数论。这些定理揭示了一个普遍的现象，即如果两个集合A和B在"测度“方面是大的，那么A加B在"序拓扑”方面就不小。建议的第二部分描述了一种计量理论方法。这种方法可以非常有效地导出一个与Shnirel'man密度或下密度定理并行的上Banach密度定理.加法数论是经典的，它研究自然数1，2，3，4.人们每天都在使用。非标准分析相对较新，也更抽象。它把无穷大的数加到我们的数字系统中。一个非专家可能会感到惊讶的是，对难以捉摸的无穷大数的研究对普通数的研究产生了有趣的影响。首席研究员在这方面已经取得了一些成功。通过进一步的努力，首席研究员将为加法数论贡献更多的创新思想和成果。

Principal Investigator: Renling JinAbstractThis research project is about applications of nonstandard analysis to additive number theory, especially to upper density or upper Banach density problems. There are many interesting theorems about Shnirel'man density or lower density in additive number theory and a few interesting theorems about upper density or upper Banach density in combinatorial number theory. However, dealing with upper density or upper Banach density in additive number theory is still an uncharted area. One of the major untouched problems in this area is finding the growth and structure of sums of sets of positive upper density or positive upper Banach density. The principal investigator has been investigating the problem using nonstandard analysis and has obtained many interesting new results. The first part of the proposal describes a topological approach to the problem. Using U-topologies on a hyperfinite Loeb space, the principal investigator proves a powerful theorem in nonstandard analysis, which implies several new theorems in some standard mathematical fields including additive number theory. These theorems reveal a general phenomenon, which says that if two sets A and B are large in terms of ``measure'', then A plus B is not small in terms of ``order-topology''. The second part of the proposal describes a measure-theoretical approach. This approach allows one to derive a parallel theorem about upper Banach density to an existing theorem about Shnirel'man density or lower density in a very efficient way.Additive number theory is classical and studies the properties of addition of natural numbers like 1,2,3,4... which people use every day. Nonstandard analysis is relatively new and more abstract. It adds infinitely large numbers into our number system. A non-expert may be surprised that the study of the elusive infinitely large numbers has an interesting impact on the study of ordinary numbers. The principal investigator has already achieved some success in this direction. By furthering his efforts, the principal investigator will contribute more innovative ideas and results to additive number theory.