New Perspectives on Generalised Lacunary Statistical Convergence of Multiset Sequences

Abstract

This paper explores the concepts of $\mathfrak{J}$-lacunary statistical limit points, $\mathfrak{J}$-lacunary statistical cluster points, and $\mathfrak{J}$-lacunary statistical Cauchy multiset sequences. Building upon previous work in the field, we investigate the relationships between $\mathfrak{J}$-lacunary statistical convergence and ${\mathfrak{J}}^{*}$-lacunary statistical convergence in multiset sequences. The findings contribute to a deeper understanding of the convergence behaviour of multiset sequences and provide new insights into the application of ideal convergence in this context.

1. Introduction

A set, according to classical set theory, is a well-defined collection of objects, where the order of elements is irrelevant and each element occurs only once. However, in some practical situations, the repetition of certain elements is meaningful. For example, a digit in a cellphone number can appear multiple times, and duplicates often arise during different phases of information retrieval. At this point, the concept of multiset becomes important. Let X represent a non-empty set. A multiset M derived from X includes elements $a\in X$ with a multiplicity function $Q\left(a\right)$, where $Q:X\to \mathbb{N}$. The positive integer $Q\left(a\right)$ signifies the multiplicity of element a. For instance, the collection $\{4,4,9,9,9,9,8,$ and $8\}$ forms a multiset since 4 appears twice, 9 appears four times, and 8 appears twice. This multiset can be represented as $\left\{4\right|2,9|4,8|2\}$, indicating that $Q\left(4\right)=2,$$Q\left(9\right)=4$, and $Q\left(8\right)=2$. Multiset theory addresses situations where classical set theory may be inadequate. In 1980, Hickman [1] investigated algebraic operations on multisets, and in 1981, Knuth examined the applications of multisets in computer programming [2]. Bender [3] focused on the partitioning of multisets, while Lake [4] provided an axiomatic framework for multiset theory in 1976. Majumdar [5] introduced soft multisets, exploring the concepts of distance and similarity between them. The theory of multisets has several applications in computer and information sciences, as discussed in references such as [3,6,7,8,9]. In 2021, Pachilangode and John [10] introduced a distance metric $d_M(a,b)=d(a,b)+\left|Q\left(a\right)-Q\left(b\right)\right|$ in a metric space $\left(X,d\right)$, and further investigated Wijsman convergence and Hausdorff convergence in relation to multisets.

The concept of statistical convergence has evolved from the idea of convergence for real number sequences, initially proposed independently by Fast [11] and Steinhaus [12], and subsequently explored by [13]. Statistical convergence is characterised by the asymptotic density of certain subsets of natural numbers. Specifically, for a subset $K\subseteq \mathbb{N}$, the asymptotic density is defined as $d\left(K\right)={lim}_{n\to \infty }{\displaystyle \frac{1}{n}}\left|\{k\le n:k\in K\}\right|$, provided this limit exists. In 1980, Šalát [14] analysed the collection of all statistically convergent sequences in ${\ell }_{\infty }$ under the supremum norm, proving that this collection is dense in ${\ell }_{\infty }$. In 1985, Fridy [15] introduced the idea of statistically Cauchy sequences and investigated the relationship between statistical convergence and these sequences. In 2000, Kostyrko et al. [16] further developed statistical convergence for real number sequences by introducing $\mathfrak{J}$-convergence, where $\mathfrak{J}$ denotes an ideal on the natural numbers $\mathbb{N}$, applicable to sequences in a metric space $\left(X,d\right)$. They also defined and studied $\mathfrak{J}$-limit points and $\mathfrak{J}$-cluster points for sequences of elements in a metric space $\left(X,d\right)$. In [17], Gürdal defined the notion of $\mathfrak{J}$-Cauchy sequences of real numbers, highlighting their connection to $\mathfrak{J}$-convergence for real number sequences (see also [18]). In this work, he also provided definitions for the $\mathfrak{J}$-Cauchy sequence and ${\mathfrak{J}}^{*}$-Cauchy sequence of elements in a metric space $(X,d)$ and established various relationships between $\mathfrak{J}$-convergence and $\mathfrak{J}$-Cauchy (see also [19]). Various papers on their associated sequence spaces may be found in [20,21,22,23,24,25,26,27,28,29,30,31].

In 2021, Debnath and Debnath [32] introduced the notion of statistical convergence specifically for multiset sequences, detailing several associated properties of this type of convergence. They also established a definition of statistical boundedness for multiset sequences and investigated the connections between statistical boundedness and statistical convergence. In 2023, Demir and Gümüş [33] focused on the concept of ideal convergence in multiset sequences.

The necessary definitions and properties for this study can be found in [10,16,19,24,32].

The purpose of this research is to explore the concept of multisets, which, unlike in classical set theory where elements are listed only once, allow for the repetition of elements in a set, with each repetition being essential. This phenomenon is commonly observed in real-world situations, highlighting the significance of multisets. Additionally, the study investigates ideal convergence, a convergence type that generalises several other forms of convergence. The aim is to understand how ideal convergence can be applied to multiset sequences and to identify the properties it offers in this context.

The framework of multisets plays an integral role in computer science and information theory, and recent years have witnessed a surge in studies investigating multiset sequences. Inspired by these developments, this work expands upon the results presented in [33]. The structure of this paper is organised as follows. Section 2 provides foundational definitions and properties related to ideals and multiset sequences. Section 3 delves into the concepts of $\mathfrak{J}$-lacunary statistical limits ($\mathfrak{JLSL}$) and $\mathfrak{J}$-lacunary statistical cluster ($\mathfrak{JLSC}$) points for multiset sequences and examines their characteristics. Additionally, we analyse the interplay between $\mathfrak{J}$-lacunary statistical convergence and $\mathfrak{J}$-lacunary statistical limit points, as well as $\mathfrak{J}$-statistical cluster points. Moreover, we establish the definitions of $\mathfrak{J}$-lacunary statistical Cauchy and ${\mathfrak{J}}^{*}$-lacunary statistical Cauchy multiset sequences. Our findings demonstrate that a multiset sequence is $\mathfrak{J}$-lacunary statistically convergent if and only if it satisfies the condition of being $\mathfrak{J}$-lacunary statistically Cauchy ($\mathfrak{JLSCauchy}$). We further show that every ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy (${\mathfrak{J}}^{*}\mathfrak{LSCauchy}$) multiset sequence is also $\mathfrak{J}$-lacunary statistically Cauchy. An illustrative example is provided to highlight the existence of ideals $\mathfrak{J}$ for which the notions of $\mathfrak{J}$-lacunary statistical Cauchy and ${\mathfrak{J}}^{*}$-lacunary statistical Cauchy are different. Furthermore, it is proven that the concepts of $\mathfrak{J}$-lacunary statistical Cauchy and ${\mathfrak{J}}^{*}$-lacunary statistical are equivalent when $\mathfrak{J}$ adheres to the weak additive property.

2. $\mathfrak{JLSL}$ and $\mathfrak{JLSC}$ Points for Multiset Sequences

Let $(Z,d)$ be a metric space. On a multiset M whose elements are from Z, one can define different types of metric; for more details, see [10]. In this paper, we consider the metric

$$d_M\left(x_n|c_n,x_m|c_m\right)=\sqrt{d{\left(x_n,x_m\right)}^2+{\left(c_n-c_m\right)}^2}$$

on M. Throughout the paper, d will denote the usual metric on $\mathbb{N}$, $d\left(x,y\right)=\left|y-x\right|$.

Definition 1

([10]). A multiset of real numbers, denoted by $m\mathbb{R}$, is a set in which the repetition of real numbers is permitted. Thus, $m\mathbb{R}=\left\{w|c:w\in \mathbb{R}\wedge c\in {\mathbb{N}}_0\right\}$.

Definition 2

([10]). A function whose domain is the set of natural numbers $\mathbb{N}$ and co-domain is $m\mathbb{R}$ is called a multiset sequence of $m\mathbb{R}$ (or, simply, a multisequence). We denote a multiset sequence by $\mathcal{G}=\left\{w_i|c_i\right\}$, where $w_i\in \mathbb{R}$ and $c_i\in \mathbb{N}$ for each $i\in \mathbb{N}$.

Example 1.

Let $N_i=\left\{1|1,2|2,\dots ,i|i\right\}$. Thus, $\left\{N_i\right\}$ represents a multiset sequence, where the $i^{nt}$ term contains ${\displaystyle \frac{i\left(i+1\right)}{2}}$ elements.

Example 2.

The integer i is fully decomposed into its prime factors. Let $T_i$ represent the multiset of these factors, including 1. For example, $T_1=\left\{1\right\}$, $T_2=\left\{1,2\right\}$, $T_3=\left\{1,3\right\}$, $T_4=\left\{1,2,2\right\}$, and $T_{36}=\left\{1,2,2,3,3\right\}$. According to this definition, $\left\{T_i\right\}$ forms a multiset sequence.

Throughout the paper, whenever we use the term multiset sequence, we will denote the multiset sequence of real numbers by $m\mathbb{R}$ unless otherwise stated. Also, for a set $\mathfrak{A}$, the complement of $\mathfrak{A}$ is represented as ${\mathfrak{A}}^c$.

By a lacunary sequence, we mean an increasing integer sequence $\theta =\left(k_u\right)$ of non-negative integers such that $k_0=0$ and $h_u=k_u-k_{u-1}$ as $u\to \infty$. Throughout this paper, the intervals determined by $\theta$ will be denoted by ${\mathcal{J}}_u=(k_{u-1},k_u]$.

A family of sets $\mathfrak{J}\subseteq 2^{\mathbb{N}}$ (power set of $\mathbb{N}$) is said to be an ideal if $\mathfrak{J}$ is additive and hereditary. A non-empty family of sets $\mathcal{F}\subseteq 2^{\mathbb{N}}$ is a filter on $\mathbb{N}$ if and only if $\varnothing \notin \mathcal{F},$ $\mathcal{A}\cap \mathcal{B}\in \mathcal{F}$ for each $\mathcal{A},\mathcal{B}\in \mathcal{F},$ and any superset of an element of $\mathcal{F}$ is in $\mathcal{F}.$ An ideal $\mathfrak{J}$ is called nontrivial if $\mathfrak{J}\ne \varnothing$ and $\mathbb{N}\notin \mathfrak{J}.$ Clearly, $\mathfrak{J}$ is a nontrivial ideal if and only if $\mathcal{F}=\mathcal{F}\left(\mathfrak{J}\right)=\left\{\mathbb{N}-\mathcal{A}:\mathcal{A}\in \mathfrak{J}\right\}$ is a filter in $\mathbb{N}$, called the filter associated with the ideal $\mathfrak{J}$. A non-trivial ideal $\mathfrak{J}$ is called admissible if and only if $\left\{\left\{n\right\}:n\in \mathbb{N}\right\}$.

We introduce the notions of $\mathfrak{J}$-lacunary statistical limit points and $\mathfrak{J}$-lacunary statistical cluster points for a sequence of multisets as follows:

Definition 3.

Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be a multiset sequence. An element $\left\{w\right|c\}$ is called an $\mathfrak{J}$-lacunary statistical limit point of $\mathfrak{H}$ if there exists a set $T=\left\{p_1<p_2<\dots <\dots \right\}\subset \mathbb{N}$ such that the set $T^{\prime }=\left\{u\in \mathbb{N}:p_u\in {\mathcal{J}}_u\right\}\notin \mathfrak{J}$ and the multiset sequence $\left\{w_{p_u}|c_{p_u}\right\}$ is lacunary statistically convergent to $w|c$. Here ${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_{\mathfrak{H}}^m\right)$ represents the collection of all $\mathfrak{J}$-lacunary statistical limit points of $\mathfrak{H}$.

Definition 4.

Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be a multiset sequence. An element $\left\{w\right|c\}$ is called an $\mathfrak{J}$-lacunary statistical cluster point of $\mathfrak{H}$ if for all $\eta >0$, $\sigma >0$:

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(w_i\left|c_i,w\right|c\right)\ge \eta \right\}\right|<\sigma \right\}\notin \mathfrak{J}.$$

Here, ${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)$ represents the collection of all $\mathfrak{J}$-lacunary statistical cluster points of $\mathfrak{H}$.

Definition 5.

A multiset sequence $\left\{w_i|c_i\right\}$ is called $\mathfrak{J}$-lacunary statistically convergent to $\left\{w\right|c\}$ if for every $\eta >0,$ $\sigma >0$:

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(w_i\left|c_i,w\right|c\right)\ge \eta \right\}\right|\ge \sigma \right\}\in \mathfrak{J},$$

and we write ${\mathfrak{J}}_{\theta }-st{lim}_{i\to \infty }{w}_i\left|c_i=w\right|c$.

Definition 6.

A multiset sequence by $\mathfrak{H}=\left\{w_i|c_i\right\}$ is called ${\mathfrak{J}}_{\theta }^{*}$-statistical convergent to $w|c$ if there exists a set $\mathfrak{M}=\left\{p_1<p_2<\cdots <p_j<\cdots \right\}\in \mathcal{F}\left(\mathfrak{J}\right)$ such that the multisubsequence $\left\{w_{p_j}|c_{p_j}\right\}$ is lacunary statistically convergent to $w|c$. It is represented as ${\mathfrak{J}}_{\theta }^{*}-st{lim}_{i\to \infty }{w}_i|c_i=w|c$.

Theorem 1.

Assume $\mathfrak{J}$ denotes any proper nontrivial admissible ideal. If ${\mathfrak{J}}_{\theta }^{*}-st{lim}_{n\to \infty }{w}_i|c_i=w|c$, then ${\mathfrak{J}}_{\theta }-st{lim}_{i\to \infty }{w}_i|c_i=w|c$.

Proof. 

Let ${\mathfrak{J}}_{\theta }^{*}-st{lim}_{i\to \infty }{w}_i|c_i=w|c$. As per the assumption, there exists a set $\mathfrak{S}\in \mathfrak{J}$ in which

$$\mathfrak{M}=\mathbb{N}\setminus \mathfrak{S}=\left\{p_1<p_2<\cdots <p_j<\cdots \right\}\in \mathcal{F}\left(\mathfrak{J}\right),$$

and we have $S_{\theta }-{lim}_{j\to \infty }{w}_{p_j}|c_{p_j}=w|c$, i.e.,

$$\underset{u\to \infty }{lim}{\displaystyle \frac{1}{h_u}}\left|\left\{p_j\in {\mathcal{J}}_u:d_M\left(w_{p_j}|c_{p_j},w|c\right)\ge \eta \right\}\right|=0.$$

Since $\mathfrak{J}$ is a proper nontrivial and admissible ideal, we deduce that

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{p_j\in {\mathcal{J}}_u:d_M\left(w_{p_j}|c_{p_j},w|c\right)\ge \eta \right\}\right|\ge \sigma \right\}\in \mathfrak{J}$$

for any $\sigma >0$. Then,

$$\begin{array}{cc}A(\eta ,\sigma )\hfill & =\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(w_i|c_i,w|c\right)\ge \eta \right\}\right|\ge \sigma \hfill \\ & \subset \mathfrak{S}\cup \left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{p_j\in {\mathcal{J}}_u:d_M\left(w_{p_j}|c_{p_j},w|c\right)\ge \eta \right\}\right|\ge \sigma \right\}\in \mathfrak{J},\hfill \end{array}$$

so, ${\mathfrak{J}}_{\theta }-st{lim}_{i\to \infty }{w}_i\left|c_i=w\right|c$. This gives the desired outcome. □

Theorem 2.

Assume $\mathfrak{J}$ denotes any proper nontrivial admissible ideal. If $\mathfrak{H}=\left\{w_i|c_i\right\}$ is a multiset sequence, then we obtain ${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_{\mathfrak{H}}^m\right)\subset {\mathfrak{J}}_{\theta }-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)$.

Proof. 

Suppose $w|c$ to be an $\mathfrak{J}$-lacunary statistical limit point of $\mathfrak{H}$. In this case, there exists a set $T=\left\{p_1<p_2<\dots <p_j<\dots \right\}\subset \mathbb{N}$ such that the set $T^{\prime }=\left\{j\in \mathbb{N}:p_j\in {\mathcal{J}}_u\right\}\notin \mathfrak{J}$ and the multiset sequence $\left\{w_{p_j}|c_{p_j}\right\}$ is lacunary statistically convergent to $w|c$. So, we have

$$\underset{u\to \infty }{lim}{\displaystyle \frac{1}{h_u}}\left|\left\{p_j\in {\mathcal{J}}_u:d_M\left(w_{p_j}|c_{p_j},w|c\right)\ge \eta \right\}\right|=0.$$

Take $\sigma >0$, so there exists $i_0\in \mathbb{N}$ such that for $u>i_0$, we have

$${\displaystyle \frac{1}{h_u}}\left|\left\{p_j\in {\mathcal{J}}_u:d_M\left(w_{p_j}|c_{p_j},w|c\right)\ge \eta \right\}\right|<\sigma .$$

Let

$$K=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{p_j\in {\mathcal{J}}_u:d_M\left(w_{p_j}|c_{p_j},w|c\right)\ge \eta \right\}\right|<\sigma \right\}.$$

In addition, we have $K\supset T\setminus \left\{p_1,p_2,\dots ,p_{i_0}\right\}$, considering that $\mathfrak{J}$ is an admissible ideal and $T\notin \mathfrak{J}$. Hence, as per to the definition of $\mathfrak{J}$-lacunary statistical cluster point, $\left\{w\right|c\}\in {\mathfrak{J}}_{\theta }-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)$. □

But, the opposite of Theorem 2 is not valid. To illustrate this, consider a mutually disjoint partition $\Im =\left\{T_1,T_2,T_3,\dots \right\}$ of $\mathbb{N}$ such that each $T_j$ is infinite. If we take

$$\mathfrak{J}=\left\{A\subset \mathbb{N}:\mathrm{there}\mathrm{exist}{l}_1,l_2,\dots ,l_p\mathrm{such}\mathrm{that}A\cap T_{l_j}\ne \varnothing \mathrm{for}\mathrm{all}j=1,2,\dots ,p\right\},$$

then $\mathfrak{J}$ becomes a proper nontrivial admissible ideal. Define a multiset sequence $\mathfrak{H}$ with same multiplicity p by

$$w_i=\left\{\begin{array}{cc}{\displaystyle \frac{1}{j}},\hfill & \mathrm{if}i\in T_j\mathrm{for}\mathrm{some}\mathrm{even}\mathrm{number}j\hfill \\ 1,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Now, consider the multiset $\left\{0\right|p\}$. Then, for any $\eta >0,$$\sigma >0,$

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(w_i|p,0|p\right)\ge \eta \right\}\right|<\sigma \right\}\notin \mathfrak{J}.$$

So, $0|p$ is an $\mathfrak{J}$-lacunary statistical cluster point of $\mathfrak{H}$. Also, it can be easily verified that $0|p$ is not an $\mathfrak{J}$-lacunary statistical limit point of $\mathfrak{H}$.

Theorem 3.

Assume $\mathfrak{H}=\left\{w_i|c_i\right\}$ is a multiset sequence and $\mathfrak{J}$ is a proper nontrivial admissible ideal on $\mathbb{N}$. Let $\mathfrak{H}$ be $\mathfrak{J}$-lacunary statistically convergent to $w|c$. Then, $w|c$ is an $\mathfrak{J}$-lacunary statistical limit point of $\mathfrak{H}$.

Proof. 

Since $\mathfrak{H}$ is $\mathfrak{J}$-lacunary statistically convergent to $w|c$, for each $\eta >0$, $\sigma >0$, the set

$$U=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{p_j\in {\mathcal{J}}_u:d_M\left(w_{p_j}|c_{p_j},w|c\right)\ge \eta \right\}\right|\ge \sigma \right\}\in \mathfrak{J},$$

where $\mathfrak{J}$ is a proper nontrivial admissible ideal on $\mathbb{N}$. Suppose that ${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_{\mathfrak{H}}^m\right)$ involves $r|c$ different from $w|c$, i.e., $r|c\in {\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_{\mathfrak{H}}^m\right)$. So, there is a $T\subset \mathbb{N}$, $T\notin \mathfrak{J}$ such that $\left\{w_{p_j}|c_{p_j}\right\}$ is lacunary statistical convergent to $r|c$. Let

$$P=\left\{u\in T:{\displaystyle \frac{1}{h_u}}\left|\left\{p_j\in {\mathcal{J}}_u:d_M\left(w_{p_j}|c_{p_j},r|c\right)\ge \eta \right\}\right|\ge \sigma \right\}.$$

Hence, P is a finite set, and so it belongs to $\mathfrak{J}$. Thus, we obtain

$$P^c=\left\{u\in T:{\displaystyle \frac{1}{h_u}}\left|\left\{p_j\in {\mathcal{J}}_u:d_M\left(w_{p_j}|c_{p_j},r|c\right)\ge \eta \right\}\right|<\sigma \right\}\in \mathcal{F}\left(\mathfrak{J}\right).$$

Again, let

$$U_1=\left\{u\in T:{\displaystyle \frac{1}{h_u}}\left|\left\{p_j\in {\mathcal{J}}_u:d_M\left(w_{p_j}|c_{p_j},w|c\right)\ge \eta \right\}\right|\ge \sigma \right\}.$$

So $U_1\subset U\in \mathfrak{J}$, i.e., $U_1^c\in \mathcal{F}\left(\mathfrak{J}\right)$. Therefore, $U_1^c\cap P^c\ne \varnothing$, since $U_1^c\cap P^c\in \mathcal{F}\left(\mathfrak{J}\right)$. Let $s\in U_1^c\cap P^c$, and take $\eta :=d_M\left(w\right|c,r\left|c\right)/3$, so ${\displaystyle \frac{1}{h_s}}\left|\left\{p_j\in {\mathcal{J}}_s:d_M\left(w_{p_j}|c_{p_j},w|c\right)\ge \eta \right\}\right|<\sigma$ and ${\displaystyle \frac{1}{h_s}}\left|\left\{p_j\in {\mathcal{J}}_s:d_M\left(w_{p_j}|c_{p_j},r|c\right)\ge \eta \right\}\right|<\sigma$, i.e., for maximum $p_j\in {\mathcal{J}}_s$ will satisfy $d_M\left(w_{p_j}|c_{p_j},w|c\right)<\eta$ and $d_M\left(w_{p_j}|c_{p_j},r|c\right)<\eta$ for a very small $\sigma >0$. Thus, we have to obtain

$$\left\{p_j\in {\mathcal{J}}_s:d_M\left(w_{p_j}\left|c_{p_j},w\right|c\right)<\eta \right\}\cap \left\{p_j\in {\mathcal{J}}_s:d_M\left(w_{p_j}\left|c_{p_j},r\right|c\right)<\eta \right\}\ne \varnothing ,$$

which is a contradiction because the neighbourhoods of $w|c$ and $r|c$ are disjoint. So, we obtain $w|c\in \mathfrak{J}-S\left({\mathrm{\Lambda }}_{\mathfrak{H}}^m\right)$. This gives the desired outcome. □

Theorem 4.

Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be a multiset sequence and $\mathfrak{J}$ be a proper nontrivial admissible ideal on $\mathbb{N}$. Let $\mathfrak{H}$ be $\mathfrak{J}$-lacunary statistically convergent to $w|c$. Then, $w|c$ is an $\mathfrak{J}$-lacunary statistical cluster point of $\mathfrak{H}$.

Proof. 

Since $\mathfrak{H}$ is $\mathfrak{J}$-lacunary statistically convergent to $w|c$, for $\eta >0$, $\sigma >0$,

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,w\right)}^2+{\left(c_i-c\right)}^2}\ge \eta \right\}\right|\ge \sigma \right\}\in \mathfrak{J}.$$

Therefore,

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\le {\mathcal{J}}_u:\sqrt{d{\left(w_i,w\right)}^2+{\left(c_i-c\right)}^2}\ge \eta \right\}\right|<\sigma \right\}\notin \mathfrak{J}.$$

Otherwise, $\mathbb{N}\in \mathfrak{J}$, which is not possible. This shows that $w|c$ is an $\mathfrak{J}$-lacunary statistical cluster point of $\mathfrak{H}$. □

Theorem 5.

For any multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$, $\mathfrak{J}-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)$ is closed.

Proof. 

Suppose $r|c$ is a limit point of the set ${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)$; then, for every $\eta >0,$${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)\cap B\left(r\right|c,\eta )\ne \varnothing$, where $B\left(r\right|c,\eta )=\left\{w|c:d_M\left(r\right|c,w\left|c\right)<\eta \right\}$. Let $q|c\in {\mathfrak{J}}_{\theta }-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)\cap \mathfrak{B}\left(r\right|c,\eta )$ and choose ${\eta }_1>0$ such that $\mathfrak{B}\left(q|c,{\eta }_1\right)\subseteq \mathfrak{B}\left(r\right|c,\eta )$. Then, we have

$$\left\{i\in {\mathcal{J}}_u:d_M\left(w_i|c_i,q|c\right)\ge {\eta }_1\right\}\supseteq \left\{i\in {\mathcal{J}}_u:d_M\left(w_i|c_i,r|c\right)\ge \eta \right\}.$$

Therefore,

$${\displaystyle \frac{1}{h_u}}\left|\{i\in {\mathcal{J}}_u:d_M\left(w_i|c_i,q|c\right)\ge {\eta }_1\right|\ge {\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(w_i|c_i,r|c\right)\ge \eta \right\}\right|.$$

For any $\sigma >0,$

$$\begin{array}{c}\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(w_i|c_i,q|c\right)\ge {\eta }_1\right\}\right|<\sigma \right\}\hfill \\ \subseteq \left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(w_i|c_i,r|c\right)\ge {\eta }_1\right\}\right|<\sigma \right\}.\hfill \end{array}$$

Since $q|c\in {\mathfrak{J}}_{\theta }-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)$, then, we have

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(w_i|c_i,r|c\right)\ge \eta \right\}\right|<\sigma \right\}\notin \mathfrak{J}.$$

Namely, $r|c\in {\mathfrak{J}}_{\theta }-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)$. Hence, the theorem is proven. □

Theorem 6.

Assume $\mathfrak{H}=\left\{w_i|c_i\right\}$ is a multiset sequence and $\mathfrak{J}$ is a proper nontrivial admissible ideal on $\mathbb{N}$. Let $\mathfrak{H}$ be ${\mathfrak{J}}_{\theta }^{*}$-lacunary statistical convergent to $w|c$. Then, $w|c$ is an ${\mathfrak{J}}_{\theta }$-lacunary statistical limit point of $\mathfrak{H}$.

Theorem 7.

Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be a multiset sequence and $\mathfrak{J},\mathcal{J}$ be two ideals on $\mathbb{N}$ with $\mathcal{J}\subset \mathfrak{J}$. If $\left\{w\right|c\}$ is an $\mathfrak{J}$-lacunary statistical limit point ($\mathfrak{J}$-lacunary statistical cluster point) of $\mathfrak{H}$, then $\left\{w\right|c\}$ is an $\mathcal{J}$-lacunary statistical limit point ($\mathcal{J}$-lacunary statistical cluster point) of $\mathfrak{H}$.

Proof. 

Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be a multiset sequence and $\mathfrak{J}$ and $\mathcal{J}$ be two ideals in $\mathbb{N}$ with $\mathcal{J}\subset \mathfrak{J}$. Let $\left\{x\right|c\}$ be an $\mathfrak{J}$-lacunary statistical limit point of $\mathfrak{H}$. Thus, there exists a set $T=\left\{p_1<p_2<\dots \right\}\notin \mathfrak{J}$ such that the multisubsequence $\left\{w_{p_j}|c_{p_j}\right\}$ is lacunary statistically convergent to $\left\{w\right|c\}$. Since $T=\left\{p_1<p_2<\dots \right\}\notin \mathcal{J}$, $\left\{w\right|c\}$ is also an $\mathcal{J}$-lacunary limit point of $\mathfrak{H}$. Similarly, we can show that $\left\{w\right|c\}$ is an $\mathcal{J}$-lacunary statistical cluster point of $\mathfrak{H}$. □

Theorem 8.

Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ and $Q=\left\{q_i|r_i\right\}$ be two multiset sequences such that

$$\left\{i\in \mathbb{N}:w_i|c_i\ne q_i|r_i\right\}\in \mathfrak{J},$$

then

1. ${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_{\mathfrak{H}}^m\right)={\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_Q^m\right)$ and

2. ${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)={\mathfrak{J}}_{\theta }-S\left({\mathrm{\Gamma }}_Q^m\right)$.

Proof. 

1. Let $w|c\in {\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_{\mathfrak{H}}^m\right)$. Hence, according to the definition, there exists a set $T=\left\{p_1<p_2<\cdots \right\}\subset \mathbb{N}$ such that $T\notin \mathfrak{J}$ and $S_{\theta }-{lim}_{j\to \infty }{w}_{p_j}|c_{p_j}=w|c$. Since

$$\left\{i\in T:w_i|c_i\ne q_i|r_i\right\}\subset \left\{i\in \mathbb{N}:w_i|c_i\ne q_i|r_i\right\}\in \mathfrak{J},$$

therefore, $T^{\prime }=\left\{i\in T:w_i|c_i=q_i|r_i\right\}\notin \mathfrak{J}$ and $T^{\prime }\subset T$. So, we have $S_{\theta }-{lim}_{i\to \infty }{q}_{p_j}|c_{p_j}=w|c$. This demonstrates that $w|c\in {\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_Q^m\right)$, and hence, ${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_{\mathfrak{H}}^m\right)\subseteq {\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_Q^m\right)$. By symmetry, ${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_Q^m\right)\subseteq {\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_{\mathfrak{H}}^m\right)$. Therefore, we obtain ${\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_{\mathfrak{H}}^m\right)={\mathfrak{J}}_{\theta }-S\left({\mathrm{\Lambda }}_Q^m\right)$.

2. Let $w|c\in \mathfrak{J}-S\left({\mathrm{\Gamma }}_{\mathfrak{H}}^m\right)$. So, by definition, for all $\eta >0$, $\sigma >0$

$$\mathfrak{U}=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(w_i|c_i,w|c\right)\ge \eta \right\}\right|<\sigma \right\}\notin \mathfrak{J}.$$

Let

$$\mathfrak{V}=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(q_i|c_i,w|c\right)\ge \eta \right\}\right|<\sigma \right\}.$$

Now, we demonstrate that $\mathfrak{V}\notin \mathfrak{J}$. First, let us assume that $\mathfrak{V}\in \mathfrak{J}$. Hence,

$${\mathfrak{V}}^c=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:d_M\left(q_i|c_i,w|c\right)\ge \eta \right\}\right|\ge \sigma \right\}\in \mathcal{F}\left(\mathfrak{J}\right).$$

By theory, the set $\mathfrak{C}=\left\{i\in \mathbb{N}:w_i|c_i=q_i|r_i\right\}$ belongs to $\mathcal{F}\left(\mathfrak{J}\right)$. Hence, ${\mathfrak{V}}^c\cap \mathfrak{C}\in \mathcal{F}\left(\mathfrak{J}\right)$. In addition, it is obvious that ${\mathfrak{V}}^c\cap \mathfrak{C}\subset {\mathfrak{A}}^c\in \mathcal{F}\left(\mathfrak{J}\right).$ This means that $\mathfrak{A}\in \mathfrak{J}$. This is a contradiction. Thus, $\mathfrak{V}\notin \mathfrak{J}$ and so the outcome is proven. □

Now, we introduce the notion of ${\mathfrak{J}}_{\theta }$-$stlimsup$ and ${\mathfrak{J}}_{\theta }$-$stliminf$ for a multiset sequence $\mathfrak{H}$ with respect to a proper nontrivial admissible ideal $\mathfrak{J}$. The introduced notion will improve the corresponding notion in [32] as the results will become a particular case for the density zero ideal ${\mathfrak{J}}_d$. We start with a multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$. Consider the two sets

$$B_{\mathfrak{H}}=\left\{w|c:\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{w_i^2+{\left(c_i-1\right)}^2}>\sqrt{w^2+{(c-1)}^2}\right\}\right|>\delta \right\}\notin \mathfrak{J}\right\}$$

and

$$A_{\mathfrak{H}}=\left\{w|c:\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{w_i^2+{\left(c_i-1\right)}^2}<\sqrt{w^2+{(c-1)}^2}\right\}\right|>\delta \right\}\notin \mathfrak{J}\right\}.$$

$\varrho =c$ is called the supremum of $B_{\mathfrak{H}}$, denoted by $sup{B}_{\mathfrak{H}}$, if c is the greatest multiplicity in $B_{\mathfrak{H}}$ under the condition $c\le max\left(c_i\right)$ in $\mathfrak{H}$ and $\varrho$ is the supremum among the different sets of real numbers bearing the multiplicity c in $B_{\mathfrak{H}}$, whenever it exists. Similarly, $inf{A}_{\mathfrak{H}}$ is defined in the same way as in [32].

Definition 7.

Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be a multiset sequence and $\mathfrak{J}$ be a proper nontrivial admissible ideal on $\mathbb{N}$. Then, we define

$$\begin{array}{cc}\hfill {\mathfrak{J}}_{\theta }-stlimsup\mathfrak{H}& =\left\{\begin{array}{c}sup{B}_{\mathfrak{H}},if{B}_{\mathfrak{H}}\ne \varphi \\ -\infty ,if{B}_{\mathfrak{H}}=\varphi \end{array}\right.\hfill \\ \hfill {\mathfrak{J}}_{\theta }-stliminf\mathfrak{H}& =\left\{\begin{array}{c}inf{A}_{\mathfrak{H}},if{A}_{\mathfrak{H}}\ne \varphi \\ +\infty ,if{A}_{\mathfrak{H}}=\varphi \end{array}\right.\hfill \end{array}$$

Example 3.

Take the ideal $\mathfrak{J}={\mathfrak{J}}_f$ containing all finite subsets of $\mathbb{N}$. Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be defined by

$$\left(w_j|c_j\right)=\left\{\begin{array}{c}0|4,ifjiseven\hfill \\ 1|3,ifjisodd.\hfill \end{array}\right.$$

Here, it can be shown that

$$B_{\mathfrak{H}}=\left\{[-2,2]\right|1,[-2,2]|2,[-2,2]|3,[-\sqrt{2},\sqrt{2}]|1,[-\sqrt{2},\sqrt{2}]\left|2\right\}.$$

So, ${\mathfrak{J}}_{\theta }-stlimsup$$\mathfrak{H}=2|3$. Also,

$$A_{\mathfrak{H}}=\{(-\infty ,-\sqrt{5}]|4,\dots ,[\sqrt{5},\infty )|4,[\sqrt{5},\infty )|5,\dots ,(-\infty ,-\sqrt{3}]|3,\dots \}$$

So, ${\mathfrak{J}}_{\theta }-stliminf\mathfrak{H}=\sqrt{5}|4$.

Theorem 9.

Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be a multiset sequence and $\mathfrak{J}$ be a nontrivial proper admissible ideal on $\mathbb{N}$. Then, ${\mathfrak{J}}_{\theta }-stlimsup\mathfrak{H}$ and ${\mathfrak{J}}_{\theta }-stliminf\mathfrak{H}$ are unique.

Proof. 

The proof is easy; thus, it has been removed. □

Theorem 10.

Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be a multiset sequence. If ${\mathfrak{J}}_{\theta }-lim\; sup\mathfrak{H}=p|r$, then, for every $\epsilon >0$,

(1) $\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{w_i^2+{\left(c_i-1\right)}^2}>\sqrt{{(p-\epsilon )}^2+{(r-1)}^2}\right\}\right|>\delta \right\}\notin \mathfrak{J}$.

(2) $\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{w_i^2+{\left(c_i-1\right)}^2}>\sqrt{{(p+\epsilon )}^2+{(r-1)}^2}\right\}\right|>\delta \right\}\in \mathfrak{J}$.

Proof. 

Since ${\mathfrak{J}}_{\theta }$-$stlimsup\mathfrak{H}=p|r$, then r is the greatest multiplicities in $B_{\mathfrak{H}}$ and p is the supremum of all real numbers whose multiplicity is r. Let $\epsilon >0$. So, there exists a real number q with $(p-\epsilon )<q$ and $q|r\in B_{\mathfrak{H}}$. Since $\sqrt{{(p-\epsilon )}^2+{(r-1)}^2}<$$\sqrt{q^2+{(r-1)}^2}$,

$$\begin{array}{c}\left\{i\in \mathbb{N}:\sqrt{w_i^2+{\left(c_i-1\right)}^2}>\sqrt{q^2+{(r-1)}^2}\right\}\hfill \\ \subset \left\{i\in \mathbb{N}:\sqrt{w_i^2+{\left(c_i-1\right)}^2}>\sqrt{{(p-\epsilon )}^2+{(r-1)}^2}\right\}.\hfill \end{array}$$

This shows that

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{w_i^2+{\left(c_i-1\right)}^2}>\sqrt{{(p-\epsilon )}^2+{(r-1)}^2}\right\}\right|>\delta \right\}\notin \mathfrak{J}.$$

On the other side, for $\epsilon >0$, if

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{w_i^2+{\left(c_i-1\right)}^2}>\sqrt{{(p+\epsilon )}^2+{(r-1)}^2}\right\}\right|>\delta \right\}\notin \mathfrak{J},$$

then $(p+\epsilon )|r\in B_{\mathfrak{H}}$, and this will contradict the fact that p is the supremum of all real numbers whose multiplicity is r. So, we have

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{w_i^2+{\left(c_i-1\right)}^2}>\sqrt{{(p+\epsilon )}^2+{(r-1)}^2}\right\}\right|>\delta \right\}\in \mathfrak{J}.$$

□

The proof of the following theorem can be given similarly to Theorem 10.

Theorem 11.

Let $\mathfrak{H}=\left\{x_i|c_i\right\}$ be a multiset sequence. If ${\mathfrak{J}}_{\theta }-stliminf\mathfrak{H}=p|r$, then, for every $\epsilon >0$,

(1) $\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{w_i^2+{\left(c_i-1\right)}^2}>\sqrt{{(p+\epsilon )}^2+{(r-1)}^2}\right\}\right|>\delta \right\}\notin \mathfrak{J}$.

(2) $\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{w_i^2+{\left(c_i-1\right)}^2}>\sqrt{{(p-\epsilon )}^2+{(r-1)}^2}\right\}\right|>\delta \right\}\in \mathfrak{J}$.

3. $\mathfrak{JLSCauchy}$ and ${\mathfrak{J}}^{*}\mathfrak{LSCauchy}$ Multiset Sequences

This section introduces and investigates $\mathfrak{J}$-lacunary and ${\mathfrak{J}}^{*}$-lacunary statistical Cauchy sequences. We demonstrate links between the ideas of $\mathfrak{J}$-lacunary statistical convergence, ${\mathfrak{J}}^{*}$-lacunary statistical convergence, $\mathfrak{J}$-lacunary statistical Cauchy, and ${\mathfrak{J}}^{*}$-lacunary statistical Cauchy multiset sequences.

Definition 8.

Let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be a multiset sequence. Then, $\mathfrak{H}=\left\{w_i|c_i\right\}$ is called $\mathfrak{J}$-lacunary statistical Cauchy if for each $\eta >0$, $\sigma >0$, there exists $N\in \mathbb{N}$ such that

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,w_N\right)}^2+{\left(c_i-c_N\right)}^2}\ge \eta \right\}\right|\ge \sigma \right\}\in \mathfrak{J}.$$

Example 4.

Let $\mathfrak{J}$ be an admissible ideal on $\mathbb{N}$ such that $A\in \mathfrak{J}$, where A is an infinite subset of $\mathbb{N}$. Let $\left\{w_i\right\}$ be any $\mathfrak{J}$-lacunary statistical Cauchy sequence of real numbers. Define a sequence of positive integers $\left\{c_i\right\}$ as follows:

$$c_i=\left\{\begin{array}{cc}i,\hfill & ifi\in A,\hfill \\ 10,\hfill & otherwise.\hfill \end{array}\right.$$

Clearly, the multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$ is $\mathfrak{J}$-lacunary statistical Cauchy.

Remark 1.

Since $\mathfrak{J}$ is an admissible ideal, every multiset Cauchy sequence is $\mathfrak{J}$-Cauchy. In particular, if ${\mathfrak{J}}_f$ is the ideal of all finite subsets of $\mathbb{N}$, then the notions of multiset Cauchy sequences and ${\mathfrak{J}}_f$-Cauchy sequences are equivalent. Let ${\mathfrak{J}}_d$ be the collection of all density-zero subsets of $\mathbb{N}$. Then, ${\mathfrak{J}}_d$ is a non-trivial admissible ideal on $\mathbb{N}$. A multiset sequence is called lacunary statistically Cauchy if and only if it is ${\mathfrak{J}}_d$-lacunary Cauchy.

Theorem 12.

A multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$ is $\mathfrak{J}$-lacunary statistically convergent if and only if it is $\mathfrak{J}$-lacunary statistical Cauchy.

Proof. 

Assume $\mathfrak{H}=\left\{w_i|c_i\right\}$ is $\mathfrak{J}$-lacunary statistically convergent to $w|c$. Then, for any $\eta ,\sigma >0$,

$$A\left({\displaystyle \frac{\eta }{2}},{\displaystyle \frac{\sigma }{2}}\right)=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,w\right)}^2+{\left(c_i-c\right)}^2}\ge {\displaystyle \frac{\eta }{2}}\right\}\right|\ge {\displaystyle \frac{\sigma }{2}}\right\}\in \mathfrak{J}.$$

Choose $N\notin A\left({\displaystyle \frac{\eta }{2}},{\displaystyle \frac{\sigma }{2}}\right)$ and fix it. Then,

$$\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_N,l\right)}^2+{\left(c_N-c\right)}^2}<{\displaystyle \frac{\eta }{2}}\right\}\right|<{\displaystyle \frac{\sigma }{2}}.$$

Also, for all $i\notin A\left({\displaystyle \frac{\eta }{2}},{\displaystyle \frac{\sigma }{2}}\right)$, we have

$$\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,l\right)}^2+{\left(c_i-c\right)}^2}<{\displaystyle \frac{\eta }{2}}\right\}\right|<{\displaystyle \frac{\sigma }{2}}.$$

Then, we have

$$\sqrt{d{\left(w_i,w_N\right)}^2+{\left(c_i-c_N\right)}^2}\le \sqrt{d{\left(w_i,l\right)}^2+{\left(c_i-c\right)}^2}+\sqrt{d{\left(w_N,l\right)}^2+{\left(c_N-c\right)}^2}<\eta ,$$

for all $i\notin A\left({\displaystyle \frac{\eta }{2}},{\displaystyle \frac{\sigma }{2}}\right)$. Consequently,

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,w_N\right)}^2+{\left(c_i-c_N\right)}^2}\ge \eta \right\}\right|\ge \sigma \right\}\subset A\left({\displaystyle \frac{\eta }{2}},{\displaystyle \frac{\sigma }{2}}\right).$$

Since $A\left({\displaystyle \frac{\eta }{2}},{\displaystyle \frac{\sigma }{2}}\right)\in \mathfrak{J}$, we have

$$\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,w_N\right)}^2+{\left(c_i-c_N\right)}^2}\ge {\displaystyle \frac{\eta }{2}}\right\}\right|\ge \sigma \right\}\in \mathfrak{J}$$

Hence, $\mathfrak{H}=\left\{w_i|c_i\right\}$ is $\mathfrak{J}$-lacunary statistical Cauchy.

Conversely, let $\mathfrak{H}=\left\{w_i|c_i\right\}$ be $\mathfrak{J}$-lacunary statistical Cauchy. Then, for any $\eta$, $\sigma >0$, there exists a positive integer $k_{\eta }$ such that

$$A(\eta ,\sigma )=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,w_{k_{\eta }}\right)}^2+{\left(c_i-c_{k_{\eta }}\right)}^2}\ge \eta \right\}\right|\ge \sigma \right\}\in \mathfrak{J}.$$

Set ${\eta }_m={\displaystyle \frac{1}{2^m}}$ for $m\in \mathbb{N}$. Hence, for all $m\in \mathbb{N}$ there exists $k_m\in \mathbb{N}$ such that

$$A(m,\sigma )=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,w_{k_m}\right)}^2+{\left(c_i-c_{k_m}\right)}^2}\ge {\displaystyle \frac{{\eta }_m}{2}}\right\}\right|\ge \sigma \right\}\in \mathfrak{J}.$$

Define recursively, $B_1=\mathrm{cl}B\left(w_{k_1},{\displaystyle \frac{{\eta }_1}{2}}\right),$ $E_1=\mathrm{cl}B\left(c_{k_1},{\displaystyle \frac{{\eta }_1}{2}}\right)$,

$$B_{m+1}=B_m\cap \mathrm{cl}B\left(w_{k_{m+1}},{\displaystyle \frac{{\eta }_{m+1}}{2}}\right),$$

and

$$E_{m+1}=E_m\cap \mathrm{cl}B\left(c_{k_{m+1}},{\displaystyle \frac{{\eta }_{m+1}}{2}}\right)$$

for $m\in \mathbb{N}$. We claim that both $B_m$ and $E_m$ are nonempty for every $m\in \mathbb{N}$. Really, since $A\left(1\right)\in \mathfrak{J}$, for each $i\notin A\left(1\right)$, we have $\sqrt{d{\left(w_i,w_{k_1}\right)}^2+{\left(c_i-c_{k_1}\right)}^2}<{\displaystyle \frac{{\eta }_1}{2}}$. Since both $d\left(w_i,w_{k_1}\right)$ and $\left|c_i-c_{k_1}\right|$ are less than or equal to $\sqrt{d{\left(w_i,w_{k_1}\right)}^2+{\left(c_i-c_{k_1}\right)}^2}$, $w_i\in B_1$ and $c_i\in E_1$. Similarly, $w_i\in \mathrm{cl}B\left(w_{k_{m+1}},{\displaystyle \frac{{\eta }_{m+1}}{2}}\right)$ and $c_i\in \mathrm{cl}B\left(c_{k_{m+1}},{\displaystyle \frac{{\eta }_{m+1}}{2}}\right)$ for all $i\notin A(m+1)$ and $m\in \mathbb{N}$. Let $m\in \mathbb{N}$ and $C\in \mathfrak{J}$ such that $w_i\in B_m$ and $c_i\in E_m$ for each $i\notin C$. Then, $w_i\in B_{m+1}$ and $c_i\in E_{m+1}$ for all $i\notin C\cup A(m+1)$. Furthermore, since $\left\{B_m\right\}$ and $\left\{E_m\right\}$ are nested sequences of closed sets of real numbers whose diameters tend to zero, there exist $l,c\in \mathbb{R}$ such that ${\bigcap }_m{B}_m=\left\{l\right\}$ and ${\cap }_m{E}_m=\left\{c\right\}$. We show that $\mathfrak{H}=\left\{w_i|c_i\right\}$ is $\mathfrak{J}$-lacunary statistically convergent to $l|c$. Let $\delta >0$ be given. So, there exists $m\in \mathbb{N}$ such that ${\eta }_m<{\displaystyle \frac{\delta }{4}}$. Let

$$B(\delta ,\sigma )=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,l\right)}^2+{\left(c_i-c\right)}^2}\ge \delta \right\}\right|\ge \sigma \right\}\in \mathfrak{J}.$$

If $B(\delta ,\sigma )$ is empty, then $B(\delta ,\sigma )\in \mathfrak{J}$. Let $B(\delta ,\sigma )\ne \varnothing$. Then, for all $i\in B(\delta ,\sigma )$, we have

$$\sqrt{d{\left(w_i,l\right)}^2+{\left(c_i-c\right)}^2}\ge \delta .$$

Since

$$\begin{array}{c}\sqrt{d{\left(w_i,l\right)}^2+{\left(c_i-c\right)}^2}\hfill \\ \le \sqrt{d{\left(w_i,w_{k_m}\right)}^2+{\left(c_i-c_{k_m}\right)}^2}+\sqrt{d{\left(l,w_{k_m}\right)}^2+{\left(c-c_{k_m}\right)}^2}\hfill \end{array},$$

either $\sqrt{d{\left(w_i,w_{k_m}\right)}^2+{\left(c_i-c_{k_m}\right)}^2}\ge {\displaystyle \frac{\delta }{2}}>2{\eta }_m$ or $\sqrt{d{\left(l,w_{k_m}\right)}^2+{\left(c-c_{k_m}\right)}^2}\ge {\displaystyle \frac{\delta }{2}}>2{\eta }_m$. But, $l\in \mathrm{cl}B\left(w_{k_m},{\displaystyle \frac{{\eta }_m}{2}}\right)$ and $c\in \mathrm{cl}B\left(c_{k_m},{\displaystyle \frac{{\eta }_m}{2}}\right)$, that is, $d\left(w_{k_m},l\right)<{\eta }_m$ and $\left|c_{k_m}-c\right|<{\eta }_m$. Therefore,

$$\sqrt{d{\left(l,w_{k_m}\right)}^2+{\left(c-c_{k_m}\right)}^2}\le d\left(w_{k_m},l\right)+\left|c_{k_m}-c\right|<2{\eta }_m.$$

Thus,

$$B(\delta ,\sigma )\subset \left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,w_{k_m}\right)}^2+{\left(c_i-c_{k_m}\right)}^2}>2{\eta }_m\ge \delta \right\}\right|\ge \sigma \right\}\subset A_m.$$

Hence the multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$ is $\mathfrak{J}$-lacunary statistically convergent to $l|c$. □

Definition 9.

A multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$ is called ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy if there exists a set $M=\left\{m_1<m_2<\cdots <m_k<\cdots \right\}\in \mathcal{F}\left(\mathfrak{J}\right)$ such that the submultiset sequence $\left\{w_{m_k}|c_{m_k}\right\}$ is lacunary statistically Cauchy.

Proposition 1.

A multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$ is ${\mathfrak{J}}^{*}$-lacunary statistically convergent if and only if it is ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy.

Theorem 13.

If a multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$ is ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy, then it is $\mathfrak{J}$-lacunary statistically Cauchy.

Proof. 

Assume $\mathfrak{H}=\left\{w_i|c_i\right\}$ is ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy. Then, there exists a set

$$M=\left\{m_1<m_2<\cdots <m_k<\cdots \right\}\in \mathcal{F}\left(\mathfrak{J}\right)$$

such that for all $\eta >0$ and for any $\sigma >0$, there exists $i_{\sigma }\in \mathbb{N}$ such that for all $k>i_{\sigma }$, we deduce that

$$\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_{m_k},w_{m_{i_{\sigma }}}\right)}^2+{\left(c_{m_k}-c_{m_{i_{\sigma }}}\right)}^2}\ge \eta \right\}\right|<\sigma .$$

Set $N=m_{i_{\sigma }}+1$ and

$$A(\eta ,\sigma )=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,w_N\right)}^2+{\left(c_i-c_N\right)}^2}\ge \eta \right\}\right|\ge \sigma \right\}.$$

Clearly, $A(\eta ,\sigma )\subset (\mathbb{N}\setminus M)\cup \left\{m_1,m_2,\cdots ,m_{i_{\sigma }}\right\}$. Since the latter set is in $\mathfrak{J}$, so does $A(\eta ,\sigma )$. Hence, $\mathfrak{H}=\left\{w_i|c_i\right\}$ is $\mathfrak{J}$-lacunary statistically Cauchy. □

The following example shows that the concepts of $\mathfrak{J}$-lacunary statistically Cauchy and ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy of multiset sequences are not generally identical:

Example 5.

Consider $\mathbb{N}={\cup }_j{A}_j$, a decomposition of $\mathbb{N}$ where each $A_j$ is infinite and $A_i\cap A_j=\varnothing$ for $i\ne j$. Let $\mathfrak{J}$ be the class of all subsets A of $\mathbb{N}$ that overlap at most a finite number of $A_j$s. It is easy to verify that $\mathfrak{J}$ is a non-trivial admissible ideal of $\mathbb{N}$. Construct a multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$ as follows: $w_i={\displaystyle \frac{1}{j}}$ if $i\in A_j$ and $c_i=10$ for all $i\in \mathbb{N}$. Let η and σ be greater than zero. Then, there is $j\in \mathbb{N}$ such that ${\displaystyle \frac{2}{j+1}}<\eta$. Let $m_0$ be the least positive integer that belongs to $A_{j+1}$. Let

$$C(\eta ,\sigma )=\left\{u\in \mathbb{N}:{\displaystyle \frac{1}{h_u}}\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(w_i,w_{m_0}\right)}^2+{\left(c_i-c_{m_0}\right)}^2}\ge \eta \right\}\right|\ge \sigma \right\}.$$

Clearly, $C(\eta ,\sigma )\subset A_1\cup A_2\cup \cdots \cup A_j$. Since the latter set belongs to $\mathfrak{J}$, so does $C(\eta ,\sigma )$. Therefore, the multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$ is $\mathfrak{J}$-lacunary statistically Cauchy.

We demonstrate that $\mathfrak{H}=\left\{w_i|c_i\right\}$ is not ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy. Suppose, on the contrary, $\mathfrak{H}=\left\{w_i|c_i\right\}$ is not ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy. Hence, there exists a set

$$M=\left\{m_1<m_2<\cdots <m_k<\cdots \right\}\in \mathcal{F}\left(\mathfrak{J}\right)$$

such that for all $\eta >0$ and for any $\sigma >0$ there exists $i_{\sigma }\in \mathbb{N}$ such that for every $k>i_{\sigma }$, we deduce that

$$\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(x_{m_k},x_{m_{i_{\sigma }}}\right)}^2+{\left(c_{m_k}-c_{m_{i_{\sigma }}}\right)}^2}\ge \eta \right\}\right|<\sigma$$

From the definition of $\mathfrak{J}$, we have $\mathbb{N}\setminus M\subset A_1\cup A_2\cup \cdots \cup A_l$ for some $l\in \mathbb{N}$. Thus, $A_i\subset M$ for all $i>l$. Therefore, there are infinitely many terms of the sequence $\left\{w_{m_k}\right\}$ equal to ${\displaystyle \frac{1}{l+1}}$ and ${\displaystyle \frac{1}{l+2}}$. Let $0<{\eta }_0<{\displaystyle \frac{1}{(l+1)(l+2)}}$. Then, there does not exist any $i_{{\eta }_0}\in \mathbb{N}$ such that

$$\left|\left\{i\in {\mathcal{J}}_u:\sqrt{d{\left(x_{m_k},x_{m_{i_{{\eta }_0}}}\right)}^2+{\left(c_{m_k}-c_{m_{i_{{\eta }_0}}}\right)}^2}\ge {\eta }_0\right\}\right|<\sigma$$

holds for all $k>i_{{\eta }_0}$. This is a contradiction. Thus, $\mathfrak{H}=\left\{w_i|c_i\right\}$ is not ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy.

Definition 10.

An admissible ideal $\mathfrak{J}$ on $\mathbb{N}$ has the weakly additive property (WAP) if for any sequence of mutually disjoint sets $\left\{P_i\right\}$ in $\mathfrak{J}$, there exists a set $P\in \mathcal{F}\left(\mathfrak{J}\right)$ such that $P\setminus P_i$ is finite for every $i\in \mathbb{N}$.

From [19] (Lemma 4), we observe that if an admissible ideal has the property (AP), it also has the property (WAP).

In order to give a sufficient requirement for a multiset $\mathfrak{J}$-lacunary statistically Cauchy sequence to be an ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy, we now add the following theorem. Also, let $\mathfrak{J}$ be an admissible ideal and have the property (WAP) in the following two results.

Theorem 14.

If a multiset sequence $\mathfrak{H}=\left\{w_i|c_i\right\}$ is $\mathfrak{J}$-lacunary statistically Cauchy, then it is ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy.

Proof. 

Assume $\mathfrak{H}=\left\{w_i|c_i\right\}$ is $\mathfrak{J}$-lacunary statistically Cauchy. Then, for all $\sigma >0$ and for any $\eta >0$, there exists a positive integer $m_{\eta }$ such that

$$A(\eta ,\sigma )=\left\{s\in \mathbb{N}:{\displaystyle \frac{1}{s}}\left|\left\{i\le s:\sqrt{d{\left(w_i,w_{m_{\eta }}\right)}^2+{\left(c_i-c_{m_{\eta }}\right)}^2}\ge \eta \right\}\right|\ge \sigma \right\}\in \mathfrak{J}.$$

Set

$$P_i=\left\{i\in \mathbb{N}:\sqrt{d{\left(w_i,w_{m_i}\right)}^2+{\left(c_i-c_{m_i}\right)}^2}<{\displaystyle \frac{1}{i}}\right\}$$

for all $i\in \mathbb{N}$. Clearly, $P_i\in \mathcal{F}\left(\mathfrak{J}\right)$ for all $i\in \mathbb{N}$. Since $\mathfrak{J}$ has the property (WAP), we have a set $P\in \mathcal{F}\left(\mathfrak{J}\right)$ such that $P\setminus P_i$ is finite for all $i\in \mathbb{N}$. Let $\delta >0$ be given. Hence, there exists $j\in \mathbb{N}$ such that ${\displaystyle \frac{1}{j}}<{\displaystyle \frac{\delta }{2}}$. Since $P\setminus P_j$ is finite, there exists a positive integer $k_j$ such that $m,i\in P_j$ for every $m,i\in P$ and $m,i>k_j$. Therefore,

$$\sqrt{d{\left(w_m,w_{m_j}\right)}^2+{\left(c_m-c_{m_j}\right)}^2}<{\displaystyle \frac{1}{j}},$$

and

$$\sqrt{d{\left(w_i,w_{m_j}\right)}^2+{\left(c_i-c_{m_j}\right)}^2}<{\displaystyle \frac{1}{j}}$$

for all $m,i\in P$ and $m,i>k_j$. Hence, by the triangle inequality of the usual norm of ${\mathbb{R}}^2$ and the metric d, we have $\sqrt{d{\left(w_m,w_i\right)}^2+{\left(c_m-c_i\right)}^2}<{\displaystyle \frac{2}{j}}<\delta$ for all $m,i\in P$ and $m,i>k_j$. With the results obtained, it is proved that $\mathfrak{H}=\left\{w_i|c_i\right\}$ is ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy. □

Corollary 1.

The notions of $\mathfrak{J}$-lacunary statistically Cauchy, $\mathfrak{J}$-statistically convergent, ${\mathfrak{J}}^{*}$-lacunary statistically Cauchy, and ${\mathfrak{J}}^{*}$-lacunary statistically convergent of multiset sequences are equivalent.

4. Discussion

In this section, we will specify the cases where $\mathfrak{J}$-statistical convergence and $\mathfrak{J}$-lacunary statistical convergence coincide with statistical and lacunary statistical convergence, as well as the types of ideals where they do not coincide.

Let ${\mathfrak{J}}_d$ (asymptotic density) and ${\mathfrak{J}}_{\delta }$ (logarithmic density) denote the class of all $A\subseteq \mathbb{N}$ for which $d\left(A\right)=0$ ($\delta \left(A\right)=0$, respectively). Then, ${\mathfrak{J}}_d$ and ${\mathfrak{J}}_{\delta }$ are non-trivial admissible ideals. ${\mathfrak{J}}_d$-convergence coincides with the statistical convergence, while ${\mathfrak{J}}_{\delta }$-convergence is referred to as logarithmic statistical convergence, as given by Kostyrko et al. [16].

Savaş and Das [20] demonstrated that a sequence is $\mathfrak{J}$-statistically convergent (according to Theorem 2.3 in [20]) but not statistically convergent (see Remark 2 in [20]) by selecting $\mathfrak{J}={\mathfrak{J}}_d$ (the ideal of density zero sets of $\mathbb{N}$). They raised an open problem, suggesting that for an arbitrary admissible ideal, $\mathfrak{J}$-lacunary statistical convergence does not coincide with lacunary statistical convergence.

In ([34], Problem 6.1), Das raised the question of identifying ideals where $\mathfrak{J}$-statistical convergence diverges from statistical convergence. Filipów and Tryba offered a partial resolution to this problem (refer to ([35], Theorem 7.16)) by employing the concept of gap density introduced by Grekos and Volkmann [36].

Definition 11

([36]). The gap density of a set $A\subseteq \mathbb{N}$ is defined as

$$\lambda \left(A\right)=\underset{u\to \infty }{lim}sup{\displaystyle \frac{e_A\left(u+1\right)}{e_A\left(u\right)}},$$

where $e_A:\mathbb{N}\to A$ represents the increasing enumeration of the set $A\subseteq \mathbb{N}$.

Filipów and Tryba [35] introduced two classes of ideals related to Das’s problem by utilising the concept of gap density.

Definition 12

([35]). An ideal $\mathfrak{J}$ is said to have property (D) if, for any M, there exists $A\in \mathfrak{F}\left(\mathfrak{J}\right)$ such that $\lambda \left(A\right)>M$.

Definition 13

([35]). An ideal $\mathfrak{J}$ is said to have property ($D^{\infty }$) if there exists $A\in \mathfrak{F}\left(\mathfrak{J}\right)$ such that $\lambda \left(A\right)=\infty$. (Sets with infinite gap density are referred to as thin sets in [37]).

The property (D) (($D^{\infty }$), resp.) serves as a necessary (or sufficient, respectively) condition for an ideal $\mathfrak{J}$ to differentiate between $\mathfrak{J}$-statistical convergence and statistical convergence.

Proposition 2

([35]). If there exists an $\mathfrak{J}$-statistically convergent sequence that is not statistically convergent, then $\mathfrak{J}$ has the property (D).

Proposition 3

([35]). If $\mathfrak{J}$ has the property ($D^{\infty }$), then there exists an $\mathfrak{J}$-statistically convergent sequence that is not statistically convergent.

An ideal $\mathfrak{J}$ is a P-ideal if for every countable family $T\subseteq \mathfrak{J}$, there is $A\in \mathfrak{J}$ such that $Q\setminus A$ is finite for every $Q\in T$.

For P-ideals, the property (D) characterises these ideals (Theorem 15), offering a partial solution to a problem posed by Das ([34], Problem 6.1).

Theorem 15

([35]). Let $\mathfrak{J}$ be a P-ideal. There exists an $\mathfrak{J}$-statistically convergent sequence that is not statistically convergent if and only if $\mathfrak{J}$ has the property (D).

Balcerzak and Leonetti [23] showed that $\mathfrak{J}$-statistical convergence coincides with $\mathcal{J}$-convergence, for some unique ideal $\mathcal{J}=\mathcal{J}\left(\mathfrak{J}\right)$, and in the same study, they proved that if $\mathfrak{J}$ is the summable ideal ${\mathfrak{J}}_{1/u}=\left\{A\subseteq \mathbb{N}:{\displaystyle \sum _{u\in A}}{\displaystyle \frac{1}{u}}<\infty \right\}$ or the density-zero ideal $\mathcal{Z}=\left\{A\subseteq \mathbb{N}:{lim}_{u\to \infty }{\displaystyle \frac{\left|A\cap \left[1,u\right]\right|}{u}}=0\right\}$, then $\mathfrak{J}$-statistical convergence coincides with statistical convergence.

Corollary 2

([23]). Let $\mathfrak{J}$ be an ideal such that $\mathfrak{J}\subseteq \mathcal{Z}$. Then, $\mathfrak{J}$-statistical convergence coincides with statistical convergence.

Moreover, as a special case of Corollary 2, they derived the following result:

Corollary 3

([23]). $\mathfrak{J}$-statistical convergence coincides with statistical convergence if $\mathfrak{J}={\mathfrak{J}}_{1/u}$ or $\mathfrak{J}=\varnothing \times$Fin, where Fin=$\left\{A\subseteq \mathbb{N}:A\mathit{is}\mathit{finite}\right\}.$

Finally, they demonstrated that the result of Corollary 2 cannot be extended to encompass the entire class of ideals $\mathfrak{J}$. In particular, this is not possible when $\mathfrak{J}$ is a maximal ideal.

Theorem 16

([23]). Let $\mathfrak{J}$ be a maximal ideal. Then, $\mathfrak{J}$-statistical convergence does not coincide with statistical convergence.

Our study is developed by considering the classes of ideals for which $\mathfrak{J}$-statistical convergence and $\mathfrak{J}$-lacunary statistical convergence do not coincide with statistical and lacunary statistical convergence, respectively. All theorem proofs are conducted in this context.

Now, using the results in the literature given above, we can say that different proofs can be obtained for our alternative main theorems, Theorem 3, Theorem 4, Theorem 9, Theorem 10, and Theorem 11. In the literature, it has been mentioned that the concepts coincide in the case of zero-density ideals. However, this is not the case for arbitrary admissible ideals, as pointed out in the works of Balcerzak-Leonetti [23], Das [34], and Savaş and Das [20]. Throughout our study, we have developed and proved our theorems mentioned above using arbitrary admissible ideals.

Therefore, in our study, $\mathfrak{J}$-statistical convergence and $\mathfrak{J}$-lacunary statistical convergence do not coincide with statistical and lacunary statistical convergence, respectively. If we take an ideal with ideal $\mathfrak{J}\subseteq \mathcal{Z}$ as in Balcerzak and Leonetti [23], our results are reduced to the results obtained with the concept of statistical convergence. On the other hand, if we take the ideal maximum instead of the ideal used in our main theorems as in [23], a different proof can be carried out by means of a submeasured sequence and results related to the $\mathfrak{J}$-statistical concept can be obtained, which again, does not coincide with statistical convergence.

Also, in the above literature, Filipów and Tryba [35] introduced two classes of ideals, defined as (D) and ($D^{\infty }$) properties, related to Das’s problem using the notion of gap density. If, instead of the ideal used in our main theorems Theorem 3, Theorem 4, Theorem 9, Theorem 10, and Theorem 11, we take the P-ideal satisfying certain conditions for each countable family and the ideal satisfying property (D), we can obtain $\mathfrak{J}$-statistical results with a different proof method that does not conflict with the statistical concept.

Considering the existing works in the above literature [23,35], we leave open questions for the interested reader to “characterise” the class of ideals $\mathfrak{J}$ for which $\mathfrak{J}$-statistical convergence coincides with statistical convergence for the class of ideals considered in our main theorems, and to determine whether the case where $\mathfrak{J}$-statistical convergence does not coincide with statistical convergence for the maximal ideal holds for unmeasurable ideals or ideals without the Baire property.

5. Conclusions

This study provides a deeper understanding of $\mathfrak{J}$-lacunary statistical limit points and $\mathfrak{J}$-lacunary statistical cluster points, as well as the behaviour of $\mathfrak{J}$-lacunary statistical Cauchy and ${\mathfrak{J}}^{*}$-lacunary statistical Cauchy multiset sequences. The equivalence of these concepts, under the condition that $\mathfrak{J}$ is an admissible ideal with the weak additive property, highlights the interconnectedness of different forms of convergence for multiset sequences. At the same time, by defining the concept of $\mathfrak{J}$-lacunary statistical convergence of multiset sequences, this work shows how this concept can be applied in different mathematical fields. In particular, the potential applications of the proposed methods in data analytics, information theory and computer science provide significant opportunities for future research. Moreover, a more comprehensive investigation of the connections with other types of convergence in multiset theory can make significant contributions to the body of knowledge in this area.