• additive combinatorics
• ergodic theory
• nonstandard methods
• Ramsey theory
• topologycal dynamics
• ultrafilters

In this thesis, we try to show how ultrafilters and nonstandard methods can be used to give a natural description of several concepts involved in the Szemeredi theorem and in its ergodic theoretical proof, as well as to give proofs of some results which are either simpler or valid in a more general setting.

In the first chapter, we recall the fundamental facts about the space of ultrafilters over a set, seen as its Stone-Cech compactification as a discrete space. In particular, in the case of a semigroup S, we describe the structure of right topological semigroup on the space of ultrafilters over S which makes it the "initial" semigroup compactification of S. In the second chapter, we define the basic notions of nonstandard methods, such as nonstandard extension, transfer principle, internal and external entities. We then recall the basic definitions and facts about nonstandard real analysis (nonstandard extensions of natural and real numbers, characterization of limit points of a net of real numbers in terms of the infinite values of its nonstandard extension, hyperfinite sets) and nonstandard measure theory (the Loeb measure associated with a finitely additive internal measure and its properties, the particular case of the counting measure).

In chapter 3, we characterize positive density sets of natural numbers, as well as other classes of sets of natural numbers of interest in Ramsey theory, in terms of their nonstandard extension with respect to a fixed nonstandard map. This characterization allows us to easily prove some facts which are not obvious from the definition, and whose standard proofs are in same cases rather intricate. Moreover, we show how some of these concepts are related to the space of ultrafilters over the set of natural numbers. In particular, we define the set of essential idempotent ultrafilters and we show that it has nice algebraic properties.

Then, we generalize these notions and characterizations to the more general setting of semigroups (chapter 4). In particular, following Hindman and Strauss, we show that in a semigroup satisfying the so called "strong Følner condition", it can be defined, in terms of Følner nets, a nicely behaving notion of density, called Følner density of the semigroup. This density has been extensively studied in the setting of countable amenable groups, where it can be defined in terms of Følner sequences rather than nets. By means of the nonstandard characterization, we extend to the uncountable case one of property, namely that if A has positive Følner density then AA^{-1} is delta-star and, in particular, syndetic. By means of this result, we generalize to the uncountable case the well known characterization of almost periodic and weakly mixing points with respect to a unitary action of a group on a Hilbert space.

In chapter 5, we show how ultrafilters can be used in topological dynamics, in particular in order to characterize recurrent and jointly recurrent points of various kinds. We hence deduce a dynamical characterization of some Ramsey-theoretical and combinatorial notions (namely, IP-sets, central and quasicentral sets, D-sets). Some of these characterizations have been given by several different authors, but in somewhat different ways (Bergelson and Downarowitz, in terms of recurrent points in a product system, Burns and Hindman, in terms of jointly recurrent points). We propose here a unifying terminology and give, for each notion, both the characterization in terms of recurrent points in a product systems and in terms of jointly recurrent points. In particular, for what concerns D-set, we extend the characterization given by Bergelson and Downarowitz to the case of a general semigroup, with respect to a density with certain properties (which are satisfied, for example by the Banach density defined by a net satisfying a condition of regularity, in a semigroup with a weak form of cancellation). We also give a proof, by means of the ultrafilter characterization, of a fact that will be used in the sequel, namely that every minimal isometric topological dynamical system is conjugated to a Kronecker system.

In chapter 7, we consider unitary actions of a group where a notion of density is defined with some characteristics (which we prove to be satisfied by the Følner density in an amenable group) on a Hilbert space, and prove, in this context, the characterization of weakly mixing and almost periodic points in terms of limits along essential idempotent ultrafilters, which is usually considered in the setting of countable amenable groups. We deduce a strong recurrence theorem for almost periodic functions which is then used in the proof of the Roth theorem. We then apply these results to measure-preserving actions on a probability space of a group with a density as above. In the following eighth chapter we consider the case of Z-actions and recall, in this context, the classical characterization of almost periodic and weakly mixing points in terms of spectral measures and limits in the sense of Cesaro and in density. By means of these characterizations and the van der Corput lemma (in its various formulations) we give a rather plain proof of the Roth theorem.

In the last part of the thesis we prove the Szemeredi theorem, following Furstenberg. First we introduce the abstract measure-preserving systems, as well as the notions of equivalence, factor and relative product of measure-preserving systems (chapter 9). Then we define and characterize almost periodic and weakly mixing extensions (chapter 10), and prove the Furstenberg-Katznelson structure theorem for measure-preserving systems. Finally, in the last chapter, we define the uniform recurrence property of a measure-preserving system and show by transfinite induction, via the structure theorem, that every measure-preserving system satisfies the uniform recurrence property. From this fact we deduce the Szemeredi theorem, as well as its multidimensional generalization for subsets of Z^{d} with positive Banach density with respect to the net of rectangles.