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

College Of Charleston, Charleston SC

Investigators

Abstract

Principal Investigator: Renling Jin Abstract This 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.