Syndetic Set Research Articles

A characterization of piecewise F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {F}$$\\end{document}-syndetic sets

Some filter relative notions of size, F,G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( \\mathscr {F},\\mathscr {G}\\right) $$\\end{document}-syndeticity and piecewise F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {F} $$\\end{document}-syndeticity, were defined and applied with clarity and focus by Shuungula, Zelenyuk and Zelenyuk (Semigroup Forum 79: 531–539, 2009). These notions are generalizations of the well studied notions of syndeticity and piecewise syndeticity. Since then, there has been an effort to develop the theory around the algebraic structure of the Stone–Čech compactification so that it encompasses these new generalizations. We prove one direction of a characterization of piecewise F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {F}$$\\end{document}-syndetic sets. This completes the characterization, as the other direction was proved by Christopherson and Johnson (Semigroup Forum 104: 28–44, 2021).

Kahane’s Upper Density and Syndetic Sets in LCA Groups

Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.

Syndetic proximal set in topological transformation group

In this paper will study the syndetic set in topological transformation group. It represents one of the most important sets in topological dynamics that Gottschak referred .We Introduce dynamical relationship between syndetic set and translation (left-right) ,In topological dynamics group present semi replet set and extensive set .We referred to the relationship of semi replet set and extensive set on the one hand, and their relationship to syndetic set on the other hand . We introduced a new concept syndetic proximal point that depended on syndetic set we indicated a relationship syndetic proximal point with proximal point and extensive proximal point , conclude a number of relationships and concepts

Force recurrence near zero

Sets that enjoy with recurrence properties near zero, in particular, sets that force recurrence near zero, sets that contain broken IP-set near zero, sets that force uniform recurrence near zero and sets that contain broken syndetic set near zero have been introduced and some relations among them are established on countable dense subsemigroup of the additive semigroup of positive real numbers. The article begins with a characterization of recurrent point near zero that affects the results exhibited here. It is also showed that how IP-set near zero can be realized dynamically.

Sets arising as minimal additive complements in the integers

A subset C of an abelian group G is said to be a minimal additive complement to W⊆G if C+W=G and if C′+W≠G for any proper subset C′⊂C. In this paper, we study certain subsets of the integers that arise or do not arise as minimal additive complements. Following a suggestion of Kwon, we show that bounded-below sets of integers with arbitrarily large gaps arise as minimal additive complements. Moreover, our construction shows that any such set belongs to a co-minimal pair, strengthening a result of Biswas and Saha for lacunary sequences in the integers. We bound the upper and lower Banach density of syndetic sets that arise as minimal additive complements to finite sets. We provide some necessary conditions for an eventually periodic set to arise as a minimal additive complement and demonstrate that these necessary conditions are also sufficient for certain classes of eventually periodic sets. We conclude with several conjectures and questions concerning the structure of minimal additive complements.

Multiplicative patterns in syndetic sets

Multiplicative patterns in syndetic sets

Hales–Jewett type configurations in small sets

In a recent work, N. Hindman, D. Strauss and L. Zamboni have shown that the Hales–Jewett theorem can be combined with a sufficiently well behaved homomorphisms. Their work was completely algebraic in nature, where they used the algebra of Stone–Čech compactification of discrete semigroups. They proved the existence of those configurations in piecewise syndetic sets, which are Ramsey theoretic rich sets. In our work we will show those forms are still present in very small but Ramsey theoretic sets, (like J-sets, C-sets) and our proof is purely elementary in nature.

Discordant sets and ergodic Ramsey theory

We explore the properties of non-piecewise syndetic sets with positive upper density, which we call "discordant", in countably infinite amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden's theorem and Szemer\'{e}di's theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.

Characterizations of relativized distal points of topological dynamical systems

Let π:(T,X)→(T,Z) be an extension of flows with phase group T. A point x in X is π-distal if x is proximal to at most itself in π−1πx∩Tx‾ under (T,X). We study the π-distal points using combinatorial methods. We present characterizations of π-distal points using product IP/Cw/C-recurrence, dynamics syndetic sets, distal sets, IP⁎-sets, and C⁎-sets in T. Moreover, we give the dynamics realization of IP-set of any discrete group by IP-recurrent point and the dynamics realization of C-set of Zd by C-recurrent point. The IP⁎-recurrence of π-distal points introduced here is useful for simplifying the proof of Furstenberg's structure theorem.

Algebraic characterizations of some relative notions of size

We obtain algebraic characterizations of relative notions of size in a discrete semigroup that generalize the usual combinatorial notions of syndetic, thick, and piecewise syndetic sets. “Filtered” syndetic and piecewise syndetic sets were defined and applied earlier by Shuungula et al. (Semigroup Forum 79:531–539, 2009). Other instances of these relative notions of size have appeared explicitly (and more often implicitly) in the literature related to the algebraic structure of the Stone–Čech compactification. Building on this prior work, we observe a natural duality and demonstrate how these notions of size may be composed to characterize previous notions of size (like piecewise syndetic sets) and serve as a convenient description for new notions of size.

Discrete quantitative Helly-type theorems with boxes

Research on Helly-type theorems in combinatorial convex geometry has produced volumetric versions of Helly's theorem using witness sets and quantitative extensions of Doignon's theorem. This paper combines these philosophies and presents quantitative Helly-type theorems for the integer lattice with axis-parallel boxes as witness sets. Our main result shows that, while quantitative Helly numbers for the integer lattice grow polynomially in each fixed dimension, their variants with boxes as witness sets are uniformly bounded. We prove several colorful and fractional variations on this theorem. We also prove that the Helly number for A×A⊆R2 need not be finite even when A⊆Z is a syndetic set.

A note on [formula omitted]-shadowing property

In this paper, we consider δ-ergodic pseudo-orbits. If any δ-pseudo-orbit can be shadowed by a point along a syndetic (aperiodic) set, then we try to connect this type of partial shadowing to the standard one. Also, a few relations on partial shadowing, not necessarily relevant, are obtained.

Geometric progressions in syndetic sets

An open problem about finite geometric progressions in syndetic sets leads to a family of diophantine equations related to the commutativity of translation and multiplication by squares.

Partition regularity and multiplicatively syndetic sets

We show how multiplicatively syndetic sets can be used in the study of partition regularity of dilation invariant systems of polynomial equations. In particular, we prove that a dilation invariant system of polynomial equations is partition regular if and only if it has a solution inside every multiplicatively syndetic set. We also adapt the methods of Green-Tao and Chow-Lindqvist-Prendiville to develop a syndetic version of Roth's density increment strategy. This argument is then used to obtain bounds on the Rado numbers of configurations of the form $\{x, d, x + d, x + 2d\}$.

Abundance of progressions in a commutative semigroup by elementary means

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k term arithmetic progression and such collection is also piecewise syndetic in Z. They used algebraic structure of beta N. The above result was extended for arbitrary semigroups by Bergelson and Hindman, again using the structure of Stone-Cech compactification of general semigroup. Beiglbock provided an elementary proof of the above result and asked whether the combinatorial argument in his proof can be enhanced in a way which makes it applicable to a more abstract setting. In this work we provide a positive answer to Beiglbock question for commutative semigroup.

Geometric progressions in syndetic sets

In order to investigate multiplicative structures in additively large sets, Beiglb\"{o}ck et al. raised a significant open question as to whether or not every subset of the natural numbers with bounded gaps (syndetic set) contains arbitrarily long geometric progressions. A result of Erd\H{o}s implies that syndetic sets contain a $2$-term geometric progression with integer common ratio, but we still do not know if they contain such a progression with common ratio being a perfect square. In this article, we prove that for each $k\in \mathbb{N}$, a syndetic set contains $2$-term geometric progressions with common ratios of the form $n^kr_1$ and $p^kr_2$, where $p\in\mathbb{P}$ (the set of primes), $n$ is a composite number, $r_1\equiv 1 \pmod{n}$, $r_2\equiv 1\pmod{p}$ and $r_1,r_2\in \mathbb{N}$. We also show that 2-syndetic sets (sets with bounded gap two) contain infinitely many $2$-term geometric progressions with their respective common ratios being perfect squares.

Large Sets in Boolean and Non-Boolean Groups and Topology

Various notions of large sets in groups, including the classical notions of thick, syndetic, and piecewise syndetic sets and the new notion of vast sets in groups, are studied with emphasis on the interplay between such sets in Boolean groups. Natural topologies closely related to vast sets are considered; as a byproduct, interesting relations between vast sets and ultrafilters are revealed.

Revisiting the nilpotent polynomial Hales–Jewett theorem

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 βG) [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.

M-systems and scattering systems of semigroup actions

We prove that: (1) an action of a semigroup S on a compact metric space X is an M-system if and only if N(x, U) is a piecewise syndetic set for every transitive point x in X and every neighborhood U of x; (2) an action of a monoid S on a compact metric space X for which every \(s \in S\) is a surjective map from X onto itself is scattering if and only if N(U, V) is a set of topological recurrence for every pair of non-empty open subsets U, V in X. As applications, we show that: (1) if an action of a commutative semigroup S on a compact metric space X is an M-system then the system is finitely sensitive; (2) an action of a commutative semigroup S on a compact metric space X is a scattering system if and only if it is disjoint with any M-system.

A multidimensional Szemerédi theorem for Hardy sequences of different growth

Abstract. We prove a variant of the multidimensional polynomial Szemeredi theorem of Bergelson and Leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some Hardy field and satisfy certain growth conditions. We do this by studying the limiting behavior of the corresponding multiple ergodic averages and obtaining a simple limit formula. A consequence of this formula in topological dynamics shows denseness of certain orbits when the iterates are restricted to suitably chosen sparse subsequences. Another consequence is that every syndetic set of integers contains certain nonshift invariant patterns, and every finite coloring of N, with each color class a syndetic set, contains certain polychromatic patterns, results very particular to our non-polynomial setup.