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Topological dynamics and combinatorial number theory
... , r − 1} by X r and let T remain the shift operator acting on X r . Specializing to our situation, we state Birkhoff's Multiple Recurrence Theorem due to Furstenberg and Weiss [11] (see also [6]). ...
... Theorem 9 (Furstenberg and Weiss [11]). Let k, r ∈ Z + . ...
This article documents my journey down the rabbit hole, chasing what I have come to know as a particularly unyielding problem in Ramsey theory on the integers: the 2-Large Conjecture. This conjecture states that if has the property that every 2-coloring of admits arbitrarily long monochromatic arithmetic progressions with common difference from D then the same property holds for any finite number of colors. We hope to provide a roadmap for future researchers and also provide some new results related to the 2-Large Conjecture.
... Using a dynamical approach, Furstenberg and Weiss [17] extended van der Waerden's theorem to arbitrary abelian groups and restricted the arithmetic structure to IP-sets. In the following definition and in the rest of this paper we use P f (X) to denote the collection of all non-empty finite subsets of a set X. ...
... Theorem 1.3 (IP van der Waerden Theorem, cf. [17,Section 3], [15,Subsection 2.5] and [22,Subsection 1.5]). Let k ∈ N, let (G, +) be an abelian group and let x 1 , . . . ...
Answering a question posed by Bergelson and Leibman in [6], we establish a nilpotent version of the polynomial Hales-Jewett theorem that contains the main theorem in [6] as a special case. Important to the formulation and the proof of our main theorem is the notion of a relative syndetic set (relative with respect to a closed non-empty subsets of ) [25]. As a corollary of our main theorem we prove an extension of the restricted van der Waerden Theorem to nilpotent groups, which involves nilprogressions.
... Poincaré's Recurrence Theorem formalizes this observation, stating that for a measure-preserving transformation T on a probability space (X , X , μ) and a set A ∈ X with μ(A) > 0, for almost any point x ∈ A there exists a sequence (n i ) i∈N such that T n i x ∈ A for all i ∈ N. The recurrence phenomenon has served to prove various deep results in different areas of mathematics. Since the works of Furstenberg [12], and Furstenberg and Weiss [13], there has been a profound and fruitful connection between dynamical systems and number theory through recurrence properties, and, more specifically, the notion of set of recurrence (see Definition 3.1) playing a central role in both areas. Within this context, the class of nilsystems has proved to be particularly important. ...
We study the topological recurrence phenomenon of actions of locally compact abelian groups on compact metric spaces. In the case of actions, we develop new techniques to analyze Bohr recurrence sets, and show that Bohr recurrence implies recurrence for -Weyl systems. This family encompasses, for example, all -affine nilsystems.
... H. Furstenberg started a systematic study of transitive systems in his paper on disjointness in topological dynamics and ergodic theory [14], and the theory was further developed in [16] and [15]. Recall that the system (X, T ) is (topologically) transitive if N T (U 1 , U 2 ) = {n ∈ Z + : U 1 ∩ T −n U 2 = ∅} (= {n ∈ Z + : T n U 1 ∩ U 2 = ∅}) ∈ P + for any opene 3 subsets U 1 , U 2 ⊂ X, equivalently, N T (U 1 , U 2 ) ∈ B for any opene subsets U 1 , U 2 ⊂ X. ...
Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in [22]. In this paper we continue to investigate this notion. In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. We investigate weak mixing and weak disjointness by using the concept of dynamical compactness. We also explore further difference between transitive compactness and weak mixing. As a byproduct, we show that the -limit and the -limit sets of a point may have quite different topological structure. Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system. Finally, these notions are also explored in the context of linear dynamics.
... Reducing Theorem 1.4 to Theorem 3.1. The elegant idea of using topological dynamics to find Ramsey families on N was developed by Furstenberg and Weiss in [21]. They considered each coloring χ : N → {1, . . . ...
An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair . We answer this question affirmatively in a strong sense by exhibiting a large new class of non-linear patterns which can be found in a single cell of any finite partition of . Our proof involves a correspondence principle which transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as and .
... 3 The Theorem in this form was spelled out in [Fur81,Theorem 8.22] but can also be deduced from [FW78,Theorem 4.4] which was published before the introduction of central sets. ...
Rado's Theorem characterizes the systems of homogenous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations. Furstenberg and Weiss proved that solutions to those systems can in fact be found in every central set. (Since one cell of any finite partition is central, this generalizes Rado's Theorem.) We show that the same holds true for the larger class of D-sets. Moreover we will see that the conclusion of Furstenberg's Central Sets Theorem is true for all sets in this class.
... The notion of IP sets, coined to abbreviate "infinite-dimensional parallelepiped" [FW78], is defined in terms of the sets FS( ) and can be considered in any semigroup. In this paper, we will always consider IP sets with respect to the additive structure (Q, +). ...
We show that every finite coloring of the rationals contains monochromatic sets of the form .
In their proof of the IP Szemer\'edi theorem, a far reaching extension of the classic theorem of Szemer\'edi on arithmetic progressions, Furstenberg and Katznelson introduced an important class of additively large sets called sets which underlies recurrence aspects in dynamics and is instrumental to enhanced formulations of combinatorial results. The authors recently showed that additive subsets of are multiplicatively rich with respect to every multiplication on without zero divisors (e.g. multiplications induced by degree d number fields). In this paper, we explain the relationships between classes of multiplicative largeness with respect to different multiplications on . We show, for example, that in contrast to the case for , there are infinitely many different notions of multiplicative piecewise syndeticity for subsets of when . This is accomplished by using the associated algebra representations to prove the existence of sets which are large with respect to some multiplications while small with respect to others. In the process, we give necessary and sufficient conditions for a linear transformation to preserve a class of multiplicatively large sets. One consequence of our results is that additive sets are multiplicatively rich in infinitely many genuinely different ways. We conclude by cataloging a number of sources of additive sets from combinatorics and dynamics.
The principal result of this paper establishes the validity of a conjecture by Graham and Rothschild. This states that, if the natural numbers are divided into two classes, then there is a sequence drawn from one of those classes such that all finite sums of distinct members of that sequence remain in the same class.