Nonstandard Methods For Upper Banach Density Problems

A general method is developed by using nonstandard analysis for formulating and proving a theorem about upper Banach density parallel to each theorem about Shnirel'man density or lower asymptotic density. There are many interesting results about Shnirel'man density or lower asymptotic density (see [4, Chapter 1] for example) in additive number theory. There are also a few interesting results about upper Banach density (see [3] or [2]) in combinatorial number theory. However, dealing with upper asymptotic density or upper Banach density in additive number theory is still an uncharted area. One of the major untouched problem in this area is nding the growth and structure of sums of sets of zero lower asymptotic density but positive upper density or upper Banach density. In this paper, we show a general method how, using nonstandard analysis, one can easily derive a parallel result about upper Banach density whenever one has a result about Shnirel'man density or lower asymptotic density in additive number theory. Serving as the testing cases of the idea, four parallel theorems about upper Banach density are formulated and proven in this paper. In x1, these four parallel theorems are stated. In x2, a brief introduction of nonstandard analysis is given. The introduction is intended for the reader without knowledge of nonstandard analysis. The reader who knows nonstandard analysis should ignore this section. In x3, all four theorems stated in x1 are proven using nonstandard analysis developed in x2. In x4, some comments are made. Mathematics Subject Classi cation Primary 11B05, 11B13, 03H05, 03H15