A nonstandard technique in combinatorial number theory. (August 2015)

• Record Type: Journal Article
• Authors: Luperi Baglini, Lorenzo
• Is Part Of: European journal of combinatorics. Volume 48(2015:Aug.)
• Page Start: 71
• Page End: 80
• Publication Date: 2015-08
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2015.02.010
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: In Di Nasso (2015) and Luperi Baglini (2012) it has been introduced a technique, based on nonstandard analysis, to study some problems in combinatorial number theory. In this paper we review such a technique and we present three of its applications: the first one is a new proof of a known result regarding the algebra of β N, namely that the center of the semigroup ( β N, ⊕ ) is N ; the second one is a generalization of a theorem of Bergelson and Hindman on arithmetic progressions of length three; the third one regards the study of which polynomials in several variables with integers coefficients have a monochromatic solution for every finite coloring of N . We will study this last application in more detail: we will prove some algebraical properties of the set P of such polynomials and we will present a few examples of nonlinear polynomials in P . In the first part of the paper we will recall the main results of the nonstandard technique that we want to use, which is based on a characterization of ultrafilters by means of nonstandard analysis.