I'm currently studying with Melvyn Nathanson,who is really considered one of the experts on additive number theory.His texts,ADDITIVE NUMBER THEORY:THE CLASSICAL BASES and ADDITIVE NUMBER THEORY:INVERSE PROBLEMS are really the standard introductions to the subject.They are both published by Springer-Verlag.
I'd also look at his papers on the Archive-he's written many of them on open problems in additive number theory.A full list can be found-with links to many of them in PDF for download-at:
http://front.math.ucdavis.edu/search?a=Nathanson%2C+Melvyn&t=&q=&c=&n=40&s=Listings.
You'll also find the most recent version of an overview of open problems in both additive number theory and combinatorics that Melvyn's been working on for a few years at:
http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.2073v1.pdf
I think you'll find the latter reference particularly pertinent to your questions.
The subject is very interesting since it essentially involves all subsets of the integers whose members can be expressed as arithmetic progressions.This provides connections to not only number theoretic questions in geometric group theory and analysis,but I'm currently investigating the role of topology in determining the structure of such "sumsets" of Z.
No comments:
Post a Comment