arXiv subjects

Sourav Kanti Patra

Publications and source records attributed to Sourav Kanti Patra.

At least 19 recordsLinked to original sources

arXiv2026

Recurrence in a dynamical system over adequate partial semigroups

Using tools from topological dynamics, H.~Furstenberg introduced the notion of \emph{central sets} and established the celebrated Central Sets Theorem. The sets which satisfy the conclusion of the Central Sets Theorem are called $C$-sets. Hindman, Maleki, and Strauss first brought the concept of an important type of $C$-sets called the quasi-central sets which are not central sets. In 2017, A. Ghosh gave the combinatorial treatment of $C$-sets in commutative adequate partial semigroups, where $C$-sets are the sets which satisfy the conclusion of Central Sets Theorem for commutative adequate partial semigroups. In this work, we discuss the Quasi-central sets algebraically and dynamically for commutative adequate partial semigroups. We give dynamical characterization of members of idempotent ultrafilters for commutative adequate partial semigroups, also we study the minimal dynamical systems for an adequate partial semigroup.

math.CO↗

arXiv2026

Ideal Zero-Product Probability in Finite Commutative Rings

We introduce the \emph{ideal zero-product probability}, a new probabilistic invariant associated with a finite commutative ring $R$, defined by \[ ζ_k(R) = \frac{ |\{(I_1,\ldots,I_k)\in\mathcal I(R)^k : I_1\cdots I_k=(0)\}| } {|\mathcal I(R)|^k}, \] where $\mathcal I(R)$ denotes the set of all ideals of $R$. This quantity gives the probability that the product of $k$ independently and uniformly chosen ideals is the zero ideal. We establish its basic properties, including invariance under ring isomorphisms, monotonicity in $k$, and multiplicativity with respect to finite direct products. Explicit formulas are obtained for finite fields, finite Boolean rings, finite chain rings, and finite principal ideal rings. For finite chain rings, we prove that the invariant depends only on the Loewy length. We also derive closed formulas using inclusion--exclusion and bounded compositions, together with generating functions. Finally, we study the asymptotic behaviour of $ζ_k(R)$ and prove that it converges to $1$ as $k\to\infty$. We also compare the ideal zero-product probability with the classical zero-product probability of ring elements on finite chain rings, obtaining explicit formulas and showing that the two invariants capture fundamentally different structural information.

math.AC↗

arXiv2026

Monochromatic Sums and Quotients Near Zero

Recently S. Goswami proved that whenever the set $\mathbb N$ of natural numbers is finitely colored, the set $\{a, b, ab, b(a+1)\}$ is monochromatic which also established a variant of the long-standing Hindman's conjecture, which asks for a monochromatic set of the form $\{a, b, ab, a+b\}$. Actually he disproved a conjecture proposed by J. Sahasrabudhe that $\{a, b, a(b + 1)\}$ is not partition regular. In this paper we prove that $\{a, b, ab, b(a+1)\}$ is monochromatic near zero which means for every finite coloring of a dense subsemigroups of $((0, \infty), +)$, the set $\{a, b, ab, b(a+1)\}$ is monochromatic near zero or in other words, we will get $a, b$ in a dense subsemigroups of $((0, \infty), +)$ as small as we want such that the set $\{a, b, ab, b(a+1)\}$ is monochromatic for every finite coloring of that dense subsemigroups of $((0, \infty), +)$, also we show that the pattern $x, y, x+y, \frac{y}{x}$ is partition regular near zero.

math.CO↗

arXiv2025

Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers

We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability $p\in (0,1)$, we prove that almost surely such a set contains finite sumsets (FS-sets) and finite product sets (FP-sets) of every finite length. In addition, we establish a novel connection between Hindman's partition theorem and the central limit theorem, providing a probabilistic perspective on the asymptotic Gaussian behavior of monochromatic finite sums and products. These results can be interpreted as probabilistic analogues of finite-dimensional versions of Hindman's theorem. Applications, implications, and open questions related to infinite FS-sets and FP-sets are discussed.

math.CO↗

arXiv2025

Multidimensional Stronger Central Sets Theorem and its Polynomial Extension

We establish and fully characterize the multidimensional extension of the Stronger Central Sets Theorem. Additionally, we develop a polynomial generalization of this result. Our approach utilizes tools from the Algebra of the Stone-Čech compactification of discrete semigroups. Several applications of these results are also discussed.

math.CO↗

arXiv2025

Graded Betti numbers of a hyperedge ideal associated to join of graphs

Let $G$ be a finite simple graph on the vertex set $V(G)$ and let $r$ be a positive integer. We consider the hypergraph $\mathrm{Con}_r(G)$ whose vertices are the vertices of $G$ and the (hyper)edges are all $A\subseteq V(G)$ such that $|A|=r+1$ and the induced subgraph $G[A]$ is connected. The (hyper)edge ideal $I_r(G)$ of $\mathrm{Con}_r(G)$ is also the Stanley-Reisner ideal of a generalisation of the independence complex of $G$, called the $r$-independence complex $\mathrm{Ind}_r(G)$. In this article we make extensive use of the Mayer-Vietoris sequence to find the graded Betti numbers of $I_r(G_1*G_2)$ in terms of the graded Betti numbers of $I_r(G_1)$ and $I_r(G_2)$, where $G_1*G_2$ is the join of $G_1$ and $G_2$. Moreover, we find formulas for all the graded Betti numbers of $I_r(G)$, when $G$ is a complete graph, complete multipartite graph, cycle graph and the wheel graph.

math.AC↗

arXiv2025

The interplay between additive and symmetric large sets and their combinatorial applications

The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic results. In this paper Di Nasso asked if his method could be adapted to find new non-linear Diophantine equations that are partition regular [7,Final remarks (4)]. By analyzing additive, multiplicative, and symmetric large sets, we construct new partition regular equations that give a first affirmative answer to this question. A special case of our result shows that if $P$ is a polynomial with no constant term then the equation $x+P(y-x)=z+w+zw$, where $y\neq x$ is partition regular. Also we prove several new monochromatic patterns involving additive, multiplicative, and symmetric structures. Throughout our work, we use tools from the Algebra of the Stone-Čech Compactifications of discrete semigroups.

math.CO↗

arXiv2024

Exponential Schur and Hindman Theorem in Ramsey Theory

Answering a conjecture of A. Sisto, J. Sahasrabudhe proved the exponential version of the Schur theorem: for every finite coloring of the naturals, there exists a monochromatic copy of $\{x,y,x^y:x\neq y\},$ which initiates the study of exponential Ramsey theory in arithmetic combinatorics. In this article, We first give two short proofs of the exponential Schur theorem, one using Zorn's lemma and another using $IP_r^\star$ van der Waerden's theorem. Then using the polynomial van der Waerden theorem iteratively we give a proof of the exponential Hindman theorem. Then applying our results we prove for every natural number $m,n$ the equation $x_n^{x_{n-1}^{\cdot^{\cdot^{\cdot^{x_1}}}}}=y_1\cdots y_m$ is partition regular, which can be considered as the exponential version of a more general version of the P. Csikvári, K. Gyarmati, and A. Sárközy conjecture, which was solved by V. Bergelson and N. Hindman independently. As a consequence of our results, we also prove that for every finite partition of $\mathbb{N},$ there exists two different sequences $\langle x_n\rangle_n$ and $\langle y_n\rangle_n$ such that both the multiplicative and exponential version of Hindman theorem generated by these sequences resp. are monochromatic, whereas in the counterpart in the finitary case, J. Sahasrabudhe proved that both sequences are the same. Our result can be considered as an exponential analog to the result of V. Bergelson and N. Hindman. We also prove that a large class of ultrafilters with certain properties do not exist, which could give us direct proof of the exponential Schur theorem. This result can be thought of as partial evidence of the nonexistence of Galvin-Glazer's proof of the exponential Hindman theorem.

math.CO↗

arXiv2024

Monocromaticity of additive and multiplicative central sets and Goswami's theorem in large Integral Domains

In \cite{Fi} A. Fish proved that if $E_{1}$ and $E_{2}$ are two subsets of $\mathbb{Z}$ of positive upper Banach density, then there exists $k\in\mathbb{Z}\setminus{0}$ such that $k\cdot\mathbb{Z}\subset\left(E_{1}-E_{1}\right)\cdot\left(E_{2}-E_{2}\right)$. In \cite{G}, S. Goswami proved the same but a fundamental result on the set of prime numbers $\mathbb{P}$ in $\mathbb{N}$ and proved that for some $k\in\mathbb{N}$, $k\cdot\mathbb{N}\subset\left(\mathbb{P}-\mathbb{P}\right)\cdot\left(\mathbb{P}-\mathbb{P}\right)$. To do so, Goswami mainly proved that the product of an $IP^{\star}$-set with an $IP_{r}^{\star}$-set contains $k\mathbb{N}$. This result is very important and surprising to mathematicians who are aware of combinatorially rich sets. In this article, we extend Goswami's result to large Lntegral Domain, that behave like $\mathbb{N}$ in the sense of some combinatorics. We prove that for a combinatorially rich ($CR$-set) set, $A$, for some $k\in\mathbb{N}$, $k\cdot\mathbb{N}\subset\left(A-A\right)\cdot\left(A-A\right)$. We provide a new proof that if we partition a large Integral Domain, $R$, into finitely many cells, then at least one cell is both additive and multiplicative central, and we prove the converse part, which is a new and unknown result.

math.CO↗

arXiv2024

Dynamical characterization of central sets in adequate partial semigroups

Using the methods from topological dynamics, H. Furstenberg introduced the notions of Central sets and proved the famous Central Sets Theorem which is the simultaneous extension of the van der Waerden and Hindman Theorem. Later N. Hindman and V. Bergelson found an equivalent formulation of Central sets in the set of natural numbers in terms of the algebra of the Stone-Čech compactification of discrete semigroups. The general case was proved by H. Shi and H. Yang. Using the notions of ultrafilters, J. McLeod introduced the notions of Central sets for commutative adequate partial semigroups, however for noncommutative cases, Central sets can be defined similarly. In this article, introducing the notions of topological dynamics for partial semigroup actions, we find an equivalent dynamical characterization of central sets in partial semigroups\footnote{Recently in \cite{GTG}, authors attempted to do the same but in a different approach.}. Throughout our article, we follow the approach of N. Hindman and D. Strauss [N. Hindman, and D. Strauss: Algebra in the Stone-\v Cech compactification: theory and applications, second edition, de Gruyter, Berlin, 2012.].

math.DS↗

arXiv2024

Product of some large sets near idempotent

We characterize when the finite Cartesian product of central sets near idempotent is central near idempotent. Moreover, we provide a partial characterization for the infinite Cartesian product of the same. Then, we study the abundance of some large sets near idempotent. Also, we investigate the effect of tensor product near idempotent. Finally, as an application we provide the polynomial extension of Milliken-Taylor theorem near zero.

math.CO↗

arXiv2024

Large sets near idempotent and its product

Tootkaboni and Vahed introduced the notion of some large sets near idempotent along with some combinatorial properties. We characterize when the finite Cartesian product of central sets near idempotent is central near idempotent. Moreover, we provide a partial characterization for the infinite Cartesian product of the same. We then study the abundance of some large sets near idempotent. We also investigate the effect of tensor product near zero. Finally, as an application we provide a characterization of members of polynomials (with constant term 0) evaluated at idempotents in a near zero semigroup.

math.CO↗

arXiv2022

Polynomial extension of the Stronger Central Sets Theorem

In [F81] Furstenberg introduced the notion of central set and established his famous Central Sets Theorem. Since then, several improved versions of Furstenberg's result have been found. The strongest generalization has been published by De, Hindman and Strauss in [DHS08], whilst a polynomial extension by Bergelson, Johnson and Moreira appeared in [BJM17]. In this article, we will establish a polynomial extension of the stronger version of the central sets theorem, and we will discuss properties of the families of sets that this result leads to consider

math.CO↗

arXiv2022

Dynamical notions along filters

We study the localization along a filter of several dynamical notions. This generalizes and extends similar localizations that have been considered in the literature, e.g. near 0 and near an idempotent. Definitions and basic properties of F-syndetic, piecewise F-syndetic, collectionwise F-piecewise syndetic, F-quasi central and F-central sets and their relations with F-uniformly recurrent points and ultrafilters are studied. We provide also the nonstandard characterizations of some of the above notions and we prove the partition regularity of several nonlinear equations along filters under mild general assumptions.

math.DS↗

arXiv2020

D-sets in Arbitrary Semigroup

We define the notion of $D$-set in an arbitrary semigroup, and with some mild restrictions we establish its dynamical and combinatorial characterizations. Assuming a weak form of cancellation in semigroups we have shown that the Cartesian product of finitely many $D$-sets is a $D$-set. A similar partial result has been proved for Cartesian product of infinitely many $D$-sets. Finally, in a commutative semigroup we deduce that $D$-sets (with respect to a Følner net) are $C$-sets.

math.DS↗

arXiv2020

Almost separable spaces

We have defined almost separable space. We show that like separability, almost separability is $c$ productive and converse also true under some restrictions. We establish a Baire Category theorem like result in Hausdorff, Pseudocompacts spaces. We investigate few relationships among separability, almost separability, sequential separability, strongly sequential separability.

math.GN↗

arXiv2019

Dynamics Near An Idempotent

Hindman and Leader first introduced the notion of semigroup of ultrafilters converging to zero for a dense subsemigroups of $((0,\infty),+)$. Using the algebraic structure of the Stone-$\breve{C}$ech compactification, Tootkabani and Vahed generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent $e$ for a dense subsemigroups of a semitopological semigroup $(T, +)$ and they gave the combinatorial proof of central set theorem near $e$. Algebraically one can also define quasi-central sets near $e$ for dense subsemigroups of $(T, +)$. In a dense subsemigroup of $(T,+)$, C-sets near $e$ are the sets, which satisfy the conclusions of the central sets theorem near $e$. S. K. Patra gave dynamical characterizations of these combinatorially rich sets near zero. In this paper we shall prove these dynamical characterizations for these combinatorially rich sets near $e$.

math.DS↗

arXiv2019

Dynamical characterization of combinatorially rich sets near zero

Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of $((0,\infty),+)$ and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-$\breve{C}$ech compactification, Bayatmanesh and Tootkabani generalized and extended this combinatorial theorem to the central theorem near zero. Algebraically one can define quasi-central set near zero for dense subsemigroup of $((0,\infty),+)$, and they also satisfy the conclusion of central sets theorem near zero. In a dense subsemigroup of $((0,\infty),+)$, C-sets near zero are the sets, which satisfies the conclusions of the central sets theorem near zero. Like discrete case, we shall produce dynamical characterizations of these combinatorically rich sets near zero.

math.DS↗