In this thesis, on a countable amenable group G, we are looking for a measure preserving system corresponding to the set E that satisfies ¯d(E) > 0. We will formulate and prove amenable version of Furstenberg correspondence principle for upper density ¯d and upper Banach density d ∗ that encompass intersections and unio‎ns of sets and their complements, and are valid for general discrete countable amenable groups. In fact, we consider a countable group G, a Folner sequence (FN ) ⊆ G and a set E ⊆ G such that ¯d(FN ) (E) = lim sup N→∞ |E ∩ FN | |FN | > 0, We show that Furstenberg correspondence principle holds, i.e; there are a measure preserving system X = (X, Σ, µ,(T g)g∈G) and a set A ∈ Σ such that µ(A) = ¯d(FN ) (E) and for every r ∈ N and every g1, · · · , gr ∈ G, we have ¯d(FN ) (g −1 1 E ∩ . . . ∩ g −1 r E) > µ((T g1) −1A ∩ . . . ∩ (T gr) −1A). (10) Then we find the ergodic measure preserving system corresponding to the set E that satisfy d ∗ (E) := sup{ ¯d(FN ) (E) : (FN )is a F olner sequence} > 0, for which statement (10) work for d ∗ instead ¯d. Then, we will pinpoint the distinction between ¯d and d ∗ which allows for the stronger, ergodic version of Furstenberg’s correspondence principle. In the following, we will give a natural generalization of Hindman’s theorem to the context of countable amenable groups. Then, we present the extension of Hindman’s covering theorem to unio‎ns of the form ∪ ∞ n=1(E − kn). In the particular, we will use ergodic theory to characterize the sequence (kn) whith the property that for any E ⊆ Z with d ∗ (E) > 0, one has d ∗ (∪ N n=1(E − kn)) → 1. Also, we introduse weakly mixing groups and provid a characterization of them, and we show that a version of Hindman’s covering theorem is valid for discrete amenable groups which are not necessarily countable.

رسول نصراصفهاني

محمود منجگاني , مهدي نعمتي