arXiv subjects

Pratulananda Das

Publications and source records attributed to Pratulananda Das.

At least 19 recordsLinked to original sources

arXiv2026

Statistically characterized subgroups related to arithmetic-type sequence of integers

Very recently, in [Das et al., J. Lond. Math. Soc., 2025], statistically characterized subgroups were studied for certain classes of non-arithmetic sequences. Subsequently, in [Das et al., Bull. Sci. Math., 2025], characterized subgroups were investigated for a class of arithmetic-type sequences that includes both arithmetic sequences and certain non-arithmetic sequences. Motivated by these developments, we study statistically characterized subgroups associated with a broader class of arithmetic-type sequences. In particular, all previously obtained cardinality related observations for statistically characterized subgroups corresponding to arithmetic sequences as well as certain non-arithmetic sequences follow as special cases of our results. Moreover, we show that this broader class exhibits drastically different behavior and differs significantly from the previously studied special cases.

math.GR↗

arXiv2025

Statistically characterized subgroups related to some non-arithmetic sequence of integers II (a quest for countable subgroups)

Following the work of [Dikranjan et al., Fund. Math. 249:185-209, 2020] for arithmetic sequences, very recently in [Das et al., Expo. Math. 43(3):125653, 2025], statistically characterized subgroups have been investigated for certain types of non-arithmetic sequences. Building on this work, we investigate further and demonstrate that, for a particular class of non-arithmetic sequences, the statistically characterized subgroup coincides with the corresponding characterized subgroup. In this context it should be kept in mind that statistical convergence (convergence w.r. to the ideal of natural density zero sets) encompasses much more sequences than usual convergence (convergence w.r. to the ideal of finite sets) and it had already been shown that statistically characterized subgroups corresponding to arithmetic sequences can not be characterized by any sequence [Das et al., Bull. Sci. Math. 179(2):103157, 2022] and they are always of the size of the continuum. From the very beginning it has been an open question as to whether statistically characterized subgroups can be small in size i.e. countably infinite. Our observation thus sheds new light on the crucial role of sequences generating subgroups of the circle group and at the same time one can subsequently identify a class of sequences for which statistically characterized subgroups are countably infinite. This result provides a negative solution to Problem 2.16 posed in [Das et al., Expo. Math. 43(3):125653, 2025] and Question 6.3 from [Dikranjan et al., Fund. Math. 249:185-209, 2020]. Additionally, our findings resolve several open problems from [Dikranjan et al., Topo. Appl., 2025].

math.GN↗

arXiv2025

Characterizing infinite torsion subgroups of the circle through arithmetic-type sequences

In a recent work [Das et al., Bull. Sci. Math. 199 (2025), 103580], the structure of characterized subgroups corresponding to arithmetic-type sequences was investigated. Building upon this work, we further show that a characterized subgroup associated with an arithmetic-type sequence is countable if and only if it is torsion. Further we prove that any infinite torsion subgroup of the circle can be characterized by an arithmetic-type sequence with bounded ratio. Moreover, our findings demonstrate that the dichotomy observed in Eggleston's theorem [Theorem 16, Eggleston, Proc. Lond. Math. Soc. 54(2) (1952), 42--93] for arithmetic sequences does not extend, in general, to the broader class of arithmetic-type sequences.

math.NT↗

arXiv2025

Certain Observations on Ideals Associated With Weighted Density Using Modulus Functions

In this article our main object of investigation is the simple modular density ideals $\mathcal{Z}_g(f)$ introduced in [Bose et al., Indag. math., 2018] where $g$ is a weight function, more precisely, $g\in G$, $G=\{g:ω\to [0,\infty):\frac{k}{g(k)}\not\to 0 \text{ and }\:\: g(k)\to \infty \text{ as }\:\:k\to \infty \}$ and $f$ is an unbounded modulus function. We mainly investigate certain properties of these ideals in line of [Kwela et al, J. math. Anal. Appl., 2019]. For an unbounded modulus function $f$ it is shown that there are $1$ or $\ck$ many functions $g\in G$ generating the same ideal $\mathcal{Z}_g(f)$. We then obtain certain interactive results involving the sequence of submeasures $\{ϕ_k\}_{k\in ω}$ generating the ideal $\mathcal{Z}_g(f)$ and the functions $g,f$. Finally, we present some observations on $\mathcal{Z}_g(f)$ ideals related to the notion of increasing-invariance.

math.GN↗

arXiv2025

A Note On Rainbow 4-Term Arithmetic Progression

Let [n]=\{1,\,2,...,\,n\} be colored in k colors. A rainbow AP(k) in [n] is a k term arithmetic progression whose elements have diferent colors. Conlon, Jungic and Radoicic [10] had shown that there exists an equinumerous 4-coloring of [4n] which happens to be rainbow AP(4) free, when n is even and subsequently Haghighi and Nowbandegani [7] shown that such a coloring of [4n] also exists when n>1 is odd. Based on their construction, we shown that a balanced 4-coloring of [n] ( i.e. size of each color class is at least \left\lfloor n/4\right\rfloor ) actually exists for all natural number n. Further we established that for nonnegative integers k\geq3 and n>1, every balanced k-coloring of [kn+r] with 0\leq r<k-1, contains a rainbow AP(k) if and only if k=3. In this paper we also have discussed about rainbow free equinumerous 4-coloring of \mathbb{Z}_{n}.

math.CO↗

arXiv2024

Statistically characterized subgroups related to some non-arithmetic sequence of integers

Recently, in Das et al. (Mediterr. J. Math. 21 : 164, 2024), characterized subgroups are investigated for some special kind of non-arithmetic sequences. In this note, we study subsequent problems in case of ``statistically characterized subgroups" introduced in Dikranjan et al. (Fund. Math. 249 : 185-209, 2020). The entire investigation emphasizes that these statistically characterized subgroups are mostly larger in size, having cardinality $\mathfrak{c}$, and exhibit behavior that significantly differs from that of classically characterized subgroups. As a consequence, we solve an open problem raised in Dikranjan et al. (Fund. Math. 249 : 185-209, 2020).

math.GR↗

arXiv2024

When ideals properly extend the class of Arbault sets

In this article we continue the investigation of generalized version of Arbault sets, that was initiated in [Das et al., Bul. Sci. Math. 179 (2022), 103157] but look at the picture from the most general point of view where ideals come into play. While Arbault sets can be naturally associated with the Frechet ideal $Fin$, in [Das et al., Bul. Sci. Math. 179 (2022), 103157] it was observed that when $Fin$ is replaced by the natural density ideal $\iI_d$ one can obtain a strictly larger class of trigonometric thin sets containing Arbault sets. From the set theoretic point of view a natural question arises as to whether one can broaden the picture and specify a class of ideals (instead of a single ideal) each of which would have the similar effect on the classical notion. As a natural candidate, we focus on a special class of ideals, namely, non-$snt$ ideals with a specific property ($snt$ stands for ``strongly non translation invariant"). This class happens to be quite large and rich as it properly contains the class of all dense translation invariant ideals ($\varsupsetneq Fin$), ideals generated by simple density functions as also certain non-negative regular summability matrices (but not all) which can be seen from [Das et al., Annals of Pure and Applied Logic 174 (2023), 103289]. We consider the resulting class of $\iI$-Arbault sets and it is observed that for each such ideal, the class of $\iI$-Arbault sets not only properly contains the class of classical Arbault sets but also a large subfamily of $\NN$-sets (also called ``sets of absolute convergence") while being contained in the class of weak Dirichlet sets.

math.GN↗

arXiv2023

On the structure of Borel ideals in-between the ideals $\mathcal{ED}$ and $\mathrm{Fin}\otimes\mathrm{Fin}$ in the Katětov order

For a family $\mathcal{F}\subseteq ω^ω$ we define the ideal $\mathcal{I}(\mathcal{F})$ on $ω\timesω$ to be the ideal generated by the family $\{A\subseteq ω\timesω:\exists f\in \mathcal{F}\,\forall^\infty n\, (|\{k:(n,k)\in A\}|\leq f(n))\}.$ Using ideals of the form $\mathcal{I}(\mathcal{F})$, we show that the structure of Borel ideals in-between two well known Borel ideals $\mathcal{ED} = \{A\subseteqω\timesω:\exists m \, \forall^\infty n\, (|\{k:(n,k)\in A\}|<m))\}$ and $\mathrm{Fin}\otimes\mathrm{Fin} = \{A\subseteqω\timesω:\forall^\infty n \, (|\{k:(n,k)\in A\}|<\aleph_0))\}$ in the Katětov order is fairly complicated. Namely, there is a copy of $\mathcal{P}(ω)/\mathrm{Fin}$ in-between $\mathcal{ED}$ and $\mathrm{Fin}\otimes\mathrm{Fin}$, and consequently there are increasing and decreasing chains of length $\mathfrak{b}$ and antichains of size $\mathfrak{c}$.

math.GN↗

arXiv2023

On certain generalized notions using $\mathcal{I}$-convergence in topological spaces

In this paper, we consider certain topological properties along with certain types of mappings on these spaces defined by the notion of ideal convergence. In order to do that, we primarily follow in the footsteps of the earlier studies of ideal convergence done by using functions (from an infinite set $S$ to $X$) in \cite{CS, das4, das5}, as that is the most general perspective and use functions instead of sequences/nets/double sequences etc. This functional approach automatically provides the most general settings for such studies and consequently extends and unifies the proofs of several old and recent results in the literature about spaces like sequential, Fréchet-Uryshon spaces and sequential, quotient and covering maps. In particular, we introduce and investigate the notions of $\ic$-functional spaces, $\ic$-functional continuous, quotient and covering mappings and finally $\ic$-functional Fréchet-Uryshon spaces. In doing so, we take help of certain set theoretic and other properties of ideals.

math.GN↗

arXiv2022

Certain observations on tightness and topological games in bornology

This article is a continuation of our investigations in the function space $C(X)$ with respect to the topology $τ^s_\mathfrak{B}$ of strong uniform convergence on $\mathfrak{B}$ in line of (Chandra et al. 2020 \cite{dcpdsd} and Das et al. 2022 \cite{pddcsd-3}) using the idea of strong uniform convergence (Beer and Levi, 2009 \cite{bl}) on a bornology. First we focus on the notion of tightness property of $(C(X),τ^s_\mathfrak{B})$ and some of its variations such as supertightness, Id-fan tightness and $T$-tightness. Certain situations are discussed when $C(X)$ is a {\rm k}-space with respect to the topology $τ^s_\mathfrak{B}$. Next the notions of strong $\mathfrak{B}$-open game and $γ_{\mathfrak{B}^s}$-open game on $X$ are introduced and some of its consequences are investigated. Finally, we consider discretely selective property and related games. On $(C(X),τ^s_\mathfrak{B})$ several interactions between topological games related to discretely selective property, the Gruenhage game on $(C(X),τ^s_\mathfrak{B})$ and certain games on $X$ are presented.

math.GN↗

arXiv2021

On certain notions of precompactness, continuity and Lipschitz functions

The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences (introduced in \cite{bc}) that has been used to define several new concepts in recent articles \cite{PDSPNA2, PDSPNA1}. We first consider a weaker notion of precompactness based on the idea of quasi-Cauchy sequences and establish several results including a new characterization of compactness in metric spaces. Next we consider associated idea of continuity, namely, ward continuous functions \cite{caka}, as this class of functions strictly lies between the classes of continuous and uniformly continuous functions and mainly establish certain coincidence results. Finally a new class of Lipschitz functions called "quasi-Cauchy Lipschitz functions" is introduced following the line of investigations in \cite{Beer1,Beer2,Beer3,g1} and again several coincidence results are proved. The motivation behind such kind of Lipschitz functions is ascertained by the observation that every real valued ward continuous function defined on a metric space can be uniformly approximated by real valued quasi-Cauchy Lipschitz functions.

math.GN↗

arXiv2020

Statistically characterized subgroups of the circle (II): continued fractions

In this note, we continue the investigation of the new version of characterized subgroups of the circle group $\mathbb{T}$, namely, "statistically characterized subgroups" (shortly, "s-characterized subgroups") recently introduced in \cite{DDB}. We primarily investigate these subgroups for sequences arising out of continued fraction representation of irrational numbers $α$ in line of \cite{L} and \cite{KL} (followed by \cite{BDMW1}) comparing their main results for this new notion and show that these subgroups are strictly larger in size (so nontrivial) than the corresponding characterized subgroups, having cardinality $\mathfrak{c}$ and containing the subgroup $\langle α\rangle$ and in the process answer the Open Question 6.4 posed in \cite{DDB}.

math.GN↗

arXiv2020

Generating Subgroups of the Circle using a Generalized class of Density Functions

In this article, we consider the generalized version $d^f_g$ of the natural density function introduced in \cite{BDK} where $g : \N \rightarrow [0,\infty)$ satisfies $g(n) \rightarrow \infty$ and $\frac{n}{g(n)} \nrightarrow 0$ whereas $f$ is an unbounded modulus function and generate versions of characterized subgroups of the circle group $\T$ using these density functions. We show that these subgroups have the same feature as the $s$-characterized subgroups \cite{DDB} or $α$-characterized subgroups \cite{BDH} and our results provide more general versions of the main results of both the articles. But at the same time the utility of this more general approach is justified by constructing new and nontrivial subgroups for suitable choice of $f$ and $g$. In several of our results we use properties of the ideal $\iZ_g(f)$ which are first presented along with certain new observations about these ideals which were not there in \cite{BDK}.

math.GN↗

arXiv2020

Topological torsion elements via natural density and a quest for solution of Armacost like problem

One can use the number theoretic idea of the notion of natural density \cite{B1} to define topological s-torsion elements (which form the statistically characterized subgroups, recently developed in \cite{DPK}) extending Armacost's idea of topological torsion elements. We follow in the line of Armacost who had posed the famous classical problem for "description of topological torsion elements" of the circle group. In this note we consider the natural density version of Armacost's problem and present a complete description of topological s-torsion elements in terms of the support, for all arithmetic sequences which also provides the solution of Problem 6.10 posed in \cite{DPK} .

math.GN↗

arXiv2020

Further observations on bornological covering properties and selection principles

This article is a continuation of the study of bornological open covers and related selection principles in metric spaces done in (Chandra et al. 2020) using the idea of strong uniform convergence (Beer and Levi, 2009) on bornology. Here we explore further ramifications, presenting characterizations of various selection principles related to certain classes of bornological covers using the Ramseyan partition relations, interactive results between the cardinalities of bornological bases and certain selection principles involving bornological covers, producing new observations on the $\mathfrak{B}^s$-Hurewicz property introduced in (Chandra et al. 2020) and several results on the $\mathfrak{B}^s$-Gerlits-Nagy property of $X$ which is introduced here following the seminal work of (Gerlits and Nagy, 1982). In addition, in the finite power $X^n$ with the product bornology $\mathfrak{B}^n$, the ${\mathfrak{B}^n}^s$-Hurewicz property as well as the ${\mathfrak{B}^n}^s$-Gerlits-Nagy property of $X^n$ are characterized in terms of properties of $(C(X),τ^s_\mathfrak{B})$ like countable fan tightness, countable strong fan tightness along with the Reznichenko's property.

math.GN↗

arXiv2019

On Leibniz algebras whose centralizers are ideals

This paper concerns the study of Leibniz algebras, a natural generalization of Lie algebras, from the perspective of centralizers of elements. We study conditions on Leibniz algebras under which centralizers of all elements are ideals. We call a Leibniz algebra, a CL-algebra if centralizers of all elements are ideals. We discuss nilpotency of CL-algebras.

math.RA↗

arXiv2019

Applications of Bornological Covering Properties in Metric Spaces

Using the idea of strong uniform convergence on bornology, Caserta, Di Maio and Kočinac studied open covers and selection principles in the realm of metric spaces (associated with a bornology) and function spaces (w.r.t. the topology of strong uniform convergence). We primarily continue in the line initiated before and investigate the behaviour of various selection principles related to these classes of bornological covers. In the process we obtain implications among these selection principles resulting in Scheepers' like diagrams. We also introduce the notion of strong-$\mathfrac{B}$-Hurewicz property and investigate some of its consequences. Finally, in $C(X)$ with respect to the topology $τ_{\mathfrac{B}}^s$ of strong uniform convergence, important properties like countable $T$-tightness, Reznichenko property are characterized in terms of bornological covering properties of $X$.

math.GN↗

arXiv2016

A notion of $αβ$-statistical convergence of order $γ$ in probability

A sequence of real numbers $\{x_{n}\}_{n\in \mathbb{N}}$ is said to be $αβ$-statistically convergent of order $γ$ (where $0<γ\leq 1$) to a real number $x$ \cite{a} if for every $δ>0,$ $$\underset{n\rightarrow \infty} {\lim} \frac{1}{(β_{n} - α_{n} + 1)^γ}~ |\{k \in [α_n,β_n] : |x_{k}-x|\geq δ\}|=0.$$ where $\{α_{n}\}_{n\in \mathbb{N}}$ and $\{β_{n}\}_{n\in \mathbb{N}}$ be two sequences of positive real numbers such that $\{α_{n}\}_{n\in \mathbb{N}}$ and $\{β_{n}\}_{n\in \mathbb{N}}$ are both non-decreasing, $β_{n}\geq α_{n}$ $\forall ~n\in \mathbb{N},$ ($β_{n}-α_{n})\rightarrow \infty$ as $n\rightarrow \infty.$ In this paper we study a related concept of convergences in which the value $|x_{k}-x|$ is replaced by $P(|X_{k}-X|\geq \varepsilon)$ and $E(|X_{k}-X|^{r})$ repectively (Where $X, X_k$ are random variables for each $k\in \mathbb{N}$, $\varepsilon>0$, $P$ denote the probability, $E$ denote the expectation) and we call them $αβ$-statistical convergence of order $γ$ in probability and $αβ$-statistical convergence of order $γ$ in $r^{\mbox{th}}$ expectation respectively. The results are applied to build the probability distribution for $αβ$-strong $p$-Ces$\grave{\mbox{a}}$ro summability of order $γ$ in probability and $αβ$-statistical convergence of order $γ$ in distribution. Our main objective is to interpret a relational behavior of above mentioned four convergences.

math.PR↗