Asymptotically lacunary ... equivalence for double set sequences defined by modulus functions

Asymptotically Lacunary

𝓘𝓘

-Equivalence for Double Set Sequences

Defined by Modulus Functions

Erdinç Dündar, Afyon Kocatepe University, Turkey, edundar@aku.edu.tr

Nimet Pancaroğlu Akın, Afyon Kocatepe University, Turkey, npancaroglu@aku.edu.tr

Abstract

Fast (1951) and Schoenberg (1959), independently, introduced concept of statistical convergence and many authors studied some properties of this concepts. Mursaleen and Edely (2003) extended this concept to the double sequences. Kostyrko et al. (2000) defined ℐ of subset of ℕ (natural numbers) and investigated convergence with some properties and proved theorems about ℐ-convergence. The idea of ℐ₂-convergence and some properties of this convergence were studied by Das et al. (2008). Nuray and Rhoades (2011) defined the idea of statistical convergence of set sequence and investigated some theorems about this notion and importance properties. After, several authors extended the convergence of real numbers sequences to convergence of sequences of sets and investigated it’s characteristic in summability.

Several authors have studied invariant convergent sequences. Tortop and Dündar (2018) introduced ℐ₂-invariant convergence of double set sequences. Asymptotically equivalent and some properties of equivalence studied by several authors. Ulusu and Gülle (2019) introduced the concept of asymptotically ℐ_𝜎𝜎-equivalence of sequences of sets. Recently, Dündar et al. (in review) studied asymptotically ideal invariant equivalence of double sequences. Many authors studied asymptotic equivalence using f modulus function. Recently, Akın, Dündar and Ulusu (2018) defined and studied asymptotically lacunary ℐ-invariant statistical equivalence for sequences of sets defined by a modulus function. Kumar and Sharma (2012) studied ℐ_𝜃𝜃-equivalent sequences using a modulus function 𝑓𝑓.

In this study, first, we present the concepts of strongly asymptotically ℐ₂𝜎𝜎𝜃𝜃-equivalence, 𝑓𝑓-asymptotically ℐ₂𝜎𝜎𝜃𝜃-equivalence, strongly 𝑓𝑓-asymptotically ℐ₂𝜎𝜎𝜃𝜃-equivalence for double sequences of sets. Then, we investigated some properties and relationships among this new concepts.

Key Words: Asymptotic Equivalence, Lacunary Invariant Convergence, ℐ₂-Convergence, Wijsman Convergence, Modulus Function.

Özet

Fast (1951) ve Schoenberg (1959) istatistiksel yakınsaklık kavramını tanımladı. Daha sonra birçok yazar bu kavram ve özellikleri ile ilgili çalışmalar yaptı. Mursaleen ve Edely (2003) istatistiksel yakınsaklık kavramını çift dizilere taşımıştır. . Kostyrko vd. (2000) ℕ’ın(doğal sayılar) alt kümesi ℐ ideal kavramını tanımladı. Sonra yeni tanımlanan ℐ yakınsaklık kavramı ile ilgili bazı özellikleri inceleyip teoremleri ispatladı. ℐ2- yakınsaklık kavramı ve bu kavramın bazı özellikleri Das vd. (2008) tarafından incelendi. Nuray

ve Rhoades (2012) küme dizileri için istatistiksel yakınsaklık kavramını tanımlayıp bu kavramla ilgili bazı özellikleri ve teoremleri inceledi. Daha sonra birçok yazar tarafından reel sayı dizilerinin yakınsaklığı küme dizilerinin yakınsaklığına genişletilerek çalışmalar yapılmıştır.

Birçok yazar invariant yakınsaklık ile ilgili çalışmalar yapmıştır. Tortop ve Dündar (2018) çift küme dizileri için ℐ2-invariant yakınsaklık kavramını tanımladı. Aşimptotik denklik ve bu denkliğin özellikleri bazı yazarlar tarafından çalışılmıştır. Ulusu ve Gülle (2019) küme dizilerinin asimptotik ℐ_𝜎𝜎-denkliği kavramını tanımladı. Son zamanlarda Dündar vd. (incelemede) çift dizilerin asimptotik ideal invariant denkliği ile ilgili çalışma yaptı. Modülüs fonksiyonu ilk defa Nakano (1953) tarafından tanımlandı. Birçok yazar f modülüs fonksiyonunu kullanarak asimptotik denklik ile ilgili çalışmalar yaptı. Akın, Dündar ve Ulusu (2018) küme dizileri için f modülüs fonksiyonunu kullanarak asimptotik lacunary ℐ-invariant istatistiksel denklik kavramını tanımlayıp bu kavram ile ilgili çalışmalar yaptı. Modülüs fonsiyonunu kullanılarak lacunary ideal denk diziler ile ilgili Kumar ve Sharma (2012) tarafından bir çalışma yapıldı.

Bu çalışmada çift küme dizileri için kuvvetli asimptotik ℐ2𝜎𝜎𝜃𝜃-denklik, , 𝑓𝑓-asimptotik ℐ2𝜎𝜎𝜃𝜃-denklik, kuvvetli 𝑓𝑓-asimptotik ℐ2𝜎𝜎𝜃𝜃

-denklik tanımları yapıldı. Daha sonra tanımlanan bu yeni kavramların özellikleri ve bu kavramlar arasındaki ilişkiler incelendi. Key Words: Asimptotik Denklik, Lacunary Invariant Yakınsaklık, ℐ₂-Yakınsaklık, Wijsman Yakınsaklık, Modülüs Fonksiyonu.

Recently, the statistical convergence extended to ideal convergence of real numbers and some important properties about ideal convergence investigated by many mathematicians. Kostyrko et al. [11] defined ℐ of subset of ℕ (positive integers) and investigated ℐ-convergence with some properties and proved theorems about ℐ-convergence. The idea of ℐ₂-convergence and some properties of this convergence were studied by Das et al. [4].

Several authors have studied invariant convergent sequences (see, [17,18, 21, 26, 27, 29, 34, 35, 36, 37, 39, 44]). Nuray et al. [22] defined the notions of invariant uniform density of subsets 𝐸𝐸 of ℕ, ℐ_𝜎𝜎-convergence and investigated relationships between ℐ_𝜎𝜎-convergence and 𝜎𝜎-convergence also ℐ_𝜎𝜎-convergence and [𝑉𝑉_𝜎𝜎]_𝑝𝑝-convergence. Tortop and Dündar [41] introduced ℐ₂-invariant convergence of double set sequences. Akın [25] studied Wijsman lacunary ℐ₂ -invariant convergence of double sequences of sets.

Several authors define some new concepts and give inclusion theorems using a modulus function 𝑓𝑓 (see, [8, 9, 14, 19, 30, 33]). Kumar and Sharma [12] studied ℐ_𝜃𝜃-equivalent sequences using a modulus function 𝑓𝑓. Kişi et al. [10] introduced 𝑓𝑓-asymptotically ℐ_𝜃𝜃-equivalent set sequences. Akın and Dündar [23] and Akın et al. [24] give definitions of 𝑓𝑓 -asymptotically ℐ_𝜎𝜎 and ℐ_{𝜎𝜎𝜃𝜃}-statistical equivalence of set sequences. Dündar and Akın [5] studied 𝑓𝑓 -asymptotically ℐ₂𝜎𝜎-equivalence for double set sequences.

Now, we recall the basic concepts and some definitions and notations (See [1, 2, 3, 10, 11, 13, 14, 15, 22, 27, 28, 31, 33, 36, 42, 43, 45, 46, 47, 48]).

Let two non-negative sequences 𝑢𝑢 = (𝑢𝑢_𝑘𝑘) and 𝑣𝑣 = (𝑣𝑣_𝑘𝑘). If lim 𝑘𝑘→∞

𝑢𝑢𝑘𝑘

𝑣𝑣𝑘𝑘= 1, then 𝑢𝑢 = (𝑢𝑢𝑘𝑘) and 𝑣𝑣 = (𝑣𝑣𝑘𝑘) are said to be

asymptotically equivalent (denoted by 𝑢𝑢~𝑣𝑣).

Let (𝑌𝑌, 𝜌𝜌) be a metric space, 𝑦𝑦 ∈ 𝑌𝑌 and any non-empty subset 𝐶𝐶 of 𝑌𝑌, then we define the distance from 𝑦𝑦 to 𝐶𝐶 by 𝑑𝑑(𝑦𝑦, 𝐶𝐶) = inf_{𝑐𝑐∈𝐶𝐶}𝜌𝜌(𝑦𝑦, 𝑐𝑐).

This after, we let (𝑌𝑌, 𝜌𝜌) be a metric space and 𝐶𝐶, 𝐷𝐷, 𝐶𝐶_𝑘𝑘 and 𝐷𝐷_𝑘𝑘 (𝑘𝑘 = 1,2, . . . ) be non-empty closed subsets of 𝑌𝑌. A sequence {𝐶𝐶_𝑘𝑘} is Wijsman convergent to 𝐶𝐶 if lim

𝑘𝑘→∞𝑑𝑑(𝑦𝑦, 𝐶𝐶𝑘𝑘) = 𝑑𝑑(𝑦𝑦, 𝐶𝐶) for each 𝑦𝑦 ∈ 𝑌𝑌. In this instance, it is showed by 𝑊𝑊 − lim

𝑘𝑘→∞𝐶𝐶𝑘𝑘= 𝐶𝐶. If sup

𝑘𝑘 𝑑𝑑(𝑦𝑦, 𝐶𝐶𝑘𝑘) < ∞ for each 𝑦𝑦 ∈ 𝑌𝑌, then {𝐶𝐶𝑘𝑘} is bounded and we write {𝐶𝐶𝑘𝑘} ∈ 𝐿𝐿∞.

Let 𝐶𝐶_𝑘𝑘, 𝐷𝐷_𝑘𝑘 ⊆ 𝑌𝑌 such that 𝑑𝑑(𝑦𝑦, 𝐶𝐶_𝑘𝑘) > 0 and 𝑑𝑑(𝑦𝑦, 𝐷𝐷_𝑘𝑘) > 0 for each 𝑦𝑦 ∈ 𝑌𝑌. The sequences {𝐶𝐶_𝑘𝑘} and {𝐷𝐷_𝑘𝑘} are asymptotically equivalent if for each 𝑦𝑦 ∈ 𝑌𝑌, lim

𝑘𝑘→∞ 𝑑𝑑(𝑦𝑦,𝐶𝐶𝑘𝑘)

𝑑𝑑(𝑦𝑦,𝐷𝐷𝑘𝑘)= 1 (denoted by 𝐶𝐶𝑘𝑘~𝐷𝐷𝑘𝑘).

Let 𝐶𝐶_𝑘𝑘, 𝐷𝐷_𝑘𝑘 ⊆ 𝑌𝑌 such that 𝑑𝑑(𝑦𝑦, 𝐶𝐶_𝑘𝑘) > 0 and 𝑑𝑑(𝑦𝑦, 𝐷𝐷_𝑘𝑘) > 0 for each 𝑦𝑦 ∈ 𝑌𝑌. If for every 𝜀𝜀 > 0 and each 𝑦𝑦 ∈ 𝑌𝑌, lim

𝑛𝑛→∞ 1

𝑛𝑛��𝑘𝑘 ≤ 𝑛𝑛: � 𝑑𝑑(𝑦𝑦,𝐶𝐶𝑘𝑘)

𝑑𝑑(𝑦𝑦,𝐷𝐷𝑘𝑘)− 𝐿𝐿� ≥ 𝜀𝜀�� = 0, then {𝐶𝐶𝑘𝑘} and {𝐷𝐷𝑘𝑘} are asymptotically statistical equivalent of multiple 𝐿𝐿

(denoted by 𝐶𝐶_𝑘𝑘𝑊𝑊𝑆𝑆~𝐿𝐿𝐷𝐷_𝑘𝑘) and if 𝐿𝐿 = 1, then {𝐶𝐶_𝑘𝑘} and{𝐷𝐷_𝑘𝑘} are asymptotically statistical equivalent.

Let 𝜎𝜎: ℕ → ℕ be a mapping and 𝜙𝜙 be a continuous linear functional on the space of real bounded sequences (ℓ_∞). 𝜙𝜙 is an invariant mean or a 𝜎𝜎-mean, if the following conditions hold:

1. 𝜙𝜙(𝑢𝑢) ≥ 0, when the sequence 𝑢𝑢 = (𝑢𝑢_𝑛𝑛) has 𝑢𝑢_𝑛𝑛≥ 0, for all 𝑛𝑛, 2. 𝜙𝜙(𝑒𝑒) = 1, where 𝑒𝑒 = (1,1,1. . . ),

3. 𝜙𝜙(𝑢𝑢_{𝜎𝜎(𝑛𝑛)}) = 𝜙𝜙(𝑢𝑢) for all 𝑢𝑢 ∈ ℓ_∞.

of 𝜎𝜎 at 𝑗𝑗. Therefore, for all 𝑢𝑢 ∈ 𝑐𝑐 𝜙𝜙(𝑢𝑢) equals to lim𝑢𝑢 which is the extension of the limit functional on 𝑐𝑐, where 𝑐𝑐 = {𝑥𝑥 = (𝑥𝑥𝑘𝑘): lim_𝑘𝑘 𝑥𝑥𝑘𝑘𝑒𝑒𝑥𝑥𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒}.

If the equality 𝜎𝜎(𝑗𝑗) ≠ 𝑗𝑗 + 1 exists, then 𝜎𝜎-mean is called a Banach limit, generally.

Now, we give the definition of ideal. ℐ ⊆ 2ℕ is called an ideal, provided that the followings hold:

(𝑒𝑒) ∅ ∈ ℐ, (𝑒𝑒𝑒𝑒) For each 𝐸𝐸, 𝐹𝐹 ∈ ℐ we have 𝐸𝐸 ∪ 𝐹𝐹 ∈ ℐ, (𝑒𝑒𝑒𝑒𝑒𝑒) For each 𝐸𝐸 ∈ ℐ and each 𝐹𝐹 ⊆ 𝐸𝐸 we have 𝐹𝐹 ∈ ℐ. Let ℐ ⊆ 2ℕ be a ideal. ℐ ⊆ 2ℕ is called non-trivial provided that ℕ ∉ ℐ. Also, for non-trivial ideal and for each 𝑛𝑛 ∈ ℕ provided that {𝑛𝑛} ∈ ℐ, then ℐ is admissible ideal. After that, we consider that ℐ is an admissible ideal.

Let 𝐶𝐶 ⊆ ℕ × ℕ and

𝑒𝑒𝑚𝑚𝑘𝑘: = min_{𝑖𝑖,𝑗𝑗} |𝐶𝐶 ∩ {(𝜎𝜎(𝑒𝑒), 𝜎𝜎(𝑗𝑗)), (𝜎𝜎2(𝑒𝑒), 𝜎𝜎2(𝑗𝑗)), . . . , (𝜎𝜎𝑚𝑚(𝑒𝑒), 𝜎𝜎𝑘𝑘(𝑗𝑗))}| and

𝑆𝑆𝑚𝑚𝑘𝑘: = max_{𝑖𝑖,𝑗𝑗} |𝐶𝐶 ∩ {(𝜎𝜎(𝑒𝑒), 𝜎𝜎(𝑗𝑗)), (𝜎𝜎2(𝑒𝑒), 𝜎𝜎2(𝑗𝑗)), . . . , (𝜎𝜎𝑚𝑚(𝑒𝑒), 𝜎𝜎𝑘𝑘(𝑗𝑗))}|.

If the limits 𝑉𝑉₂(𝐶𝐶): = lim 𝑚𝑚,𝑘𝑘→∞

𝑠𝑠𝑚𝑚𝑘𝑘

𝑚𝑚𝑘𝑘 and 𝑉𝑉2(𝐶𝐶): = lim𝑚𝑚,𝑘𝑘→∞ 𝑆𝑆𝑚𝑚𝑘𝑘

𝑚𝑚𝑘𝑘 exists then 𝑉𝑉2(𝐶𝐶) is called a lower and 𝑉𝑉2(𝐶𝐶) is called an upper 𝜎𝜎-uniform density of the set 𝐶𝐶, respectively. If 𝑉𝑉₂(𝐶𝐶) = 𝑉𝑉₂(𝐶𝐶), then 𝑉𝑉₂(𝐶𝐶) = 𝑉𝑉₂(𝐶𝐶) = 𝑉𝑉₂(𝐶𝐶) is called the 𝜎𝜎-uniform density of 𝐶𝐶. Denote by ℐ₂𝜎𝜎 the class of all 𝐶𝐶 ⊆ ℕ × ℕ with 𝑉𝑉₂(𝐶𝐶) = 0.

This after, let 𝐶𝐶_{𝑖𝑖𝑗𝑗}, 𝐷𝐷_{𝑖𝑖𝑗𝑗}, 𝐶𝐶, 𝐷𝐷 be any nonempty closed subsets of 𝑌𝑌. If for each 𝑦𝑦 ∈ 𝑌𝑌, lim 𝑚𝑚,𝑘𝑘→∞ 1 𝑚𝑚𝑘𝑘 � 𝑚𝑚,𝑘𝑘 𝑖𝑖,𝑗𝑗=1,1 𝑑𝑑(𝑦𝑦, 𝐶𝐶_𝜎𝜎𝑖𝑖_{(𝑠𝑠),𝜎𝜎}𝑗𝑗_(𝑡𝑡)) = 𝑑𝑑(𝑦𝑦, 𝐶𝐶),

uniformly in 𝑒𝑒, 𝑒𝑒 then, the double sequence {𝐶𝐶𝑖𝑖𝑗𝑗} is said to be invariant convergent to 𝐶𝐶 in 𝑌𝑌. If for every 𝛾𝛾 > 0,

𝐴𝐴(𝛾𝛾, 𝑦𝑦) = {(𝑒𝑒, 𝑗𝑗): |𝑑𝑑(𝑦𝑦, 𝐶𝐶𝑖𝑖𝑗𝑗) − 𝑑𝑑(𝑦𝑦, 𝐶𝐶)| ≥ 𝛾𝛾} ∈ ℐ2𝜎𝜎

that is, 𝑉𝑉₂(𝐴𝐴(𝛾𝛾, 𝑦𝑦)) = 0 then, the double sequence {𝐶𝐶_{𝑖𝑖𝑗𝑗}} is said to be Wijsman ℐ₂-invariant convergent or ℐ_𝑊𝑊𝜎𝜎₂ -convergent to 𝐶𝐶, In this instance, we write 𝐶𝐶_{𝑖𝑖𝑗𝑗} → 𝐶𝐶(ℐ_𝑊𝑊𝜎𝜎₂) and by ℐ_𝑊𝑊𝜎𝜎₂ we will denote the set of all Wijsman ℐ₂𝜎𝜎 -convergent double sequences of sets.

For non-empty closed subsets 𝐶𝐶_{𝑖𝑖𝑗𝑗}, 𝐷𝐷_{𝑖𝑖𝑗𝑗} of 𝑌𝑌 define 𝑑𝑑(𝑦𝑦; 𝐶𝐶_{𝑖𝑖𝑗𝑗}, 𝐷𝐷_{𝑖𝑖𝑗𝑗}) as follows:

𝑑𝑑(𝑦𝑦; 𝐶𝐶𝑖𝑖𝑗𝑗, 𝐷𝐷𝑖𝑖𝑗𝑗) = � 𝑑𝑑(𝑦𝑦,𝐶𝐶𝑖𝑖𝑗𝑗)

𝑑𝑑(𝑦𝑦,𝐷𝐷𝑖𝑖𝑗𝑗) , 𝑦𝑦 ∈ 𝐶𝐶𝑖𝑖𝑗𝑗∪ 𝐷𝐷𝑖𝑖𝑗𝑗

𝐿𝐿 , 𝑦𝑦 ∈ 𝐶𝐶𝑖𝑖𝑗𝑗∪ 𝐷𝐷𝑖𝑖𝑗𝑗. If for every 𝛾𝛾 > 0 and for each 𝑦𝑦 ∈ 𝑌𝑌,

�(𝑚𝑚, 𝑘𝑘): ∈ ℕ × ℕ:_{𝑚𝑚𝑘𝑘 �}1 𝑚𝑚,𝑘𝑘 𝑖𝑖,𝑗𝑗=1,1

|𝑑𝑑(𝑦𝑦; 𝐶𝐶𝑖𝑖𝑗𝑗, 𝐷𝐷𝑖𝑖𝑗𝑗) − 𝐿𝐿| ≥ 𝛾𝛾� ∈ ℐ2𝜎𝜎

then, double sequences {𝐶𝐶_{𝑖𝑖𝑗𝑗}} and {𝐷𝐷_{𝑖𝑖𝑗𝑗}} are said to be strongly asymptotically ℐ₂𝜎𝜎-equivalent of multiple 𝐿𝐿 (denoted by 𝐶𝐶_{𝑖𝑖𝑗𝑗} ~

[𝑊𝑊_ℐ2𝜎𝜎𝐿𝐿]

𝐷𝐷𝑖𝑖𝑗𝑗) and if 𝐿𝐿 = 1, then {𝐶𝐶𝑖𝑖𝑗𝑗} and {𝐷𝐷𝑖𝑖𝑗𝑗} are said to be strongly asymptotically ℐ2𝜎𝜎-equivalent. If following conditions hold for 𝑓𝑓: [0, ∞) → [0, ∞) function, then it is called a modulus function:

1. 𝑓𝑓(𝑢𝑢) = 0 if and if only if 𝑢𝑢 = 0, 2. 𝑓𝑓(𝑢𝑢 + 𝑣𝑣) ≤ 𝑓𝑓(𝑢𝑢) + 𝑓𝑓(𝑣𝑣) 3. 𝑓𝑓 is increasing

4. 𝑓𝑓 is continuous from the right at 0. This after, we let 𝑓𝑓 as a modulus function.

The modulus function 𝑓𝑓 may be unbounded (for instance 𝑓𝑓(𝑢𝑢) = 𝑢𝑢𝑞𝑞, 0 < 𝑞𝑞 < 1) or bounded (for example 𝑓𝑓(𝑢𝑢) = 𝑢𝑢

𝑢𝑢+1).

If for every 𝛾𝛾 > 0 and for each 𝑦𝑦 ∈ 𝑌𝑌,

{(𝑒𝑒, 𝑗𝑗) ∈ ℕ × ℕ: 𝑓𝑓(|𝑑𝑑(𝑦𝑦; 𝐶𝐶𝑖𝑖𝑗𝑗, 𝐷𝐷𝑖𝑖𝑗𝑗) − 𝐿𝐿|) ≥ 𝛾𝛾} ∈ ℐ2𝜎𝜎,

then the double sequences {𝐶𝐶_{𝑖𝑖𝑗𝑗}} and {𝐷𝐷_{𝑖𝑖𝑗𝑗}} are said to be 𝑓𝑓-asymptotically ℐ₂𝜎𝜎-equivalent of multiple 𝐿𝐿 (denoted by 𝐶𝐶𝑖𝑖𝑗𝑗 ~

𝑊𝑊_ℐ2𝜎𝜎𝐿𝐿(𝑓𝑓)

𝐷𝐷𝑖𝑖𝑗𝑗) and if 𝐿𝐿 = 1, then {𝐶𝐶𝑖𝑖𝑗𝑗} and {𝐷𝐷𝑖𝑖𝑗𝑗} are said to be 𝑓𝑓-asymptotically ℐ2𝜎𝜎-equivalent. If for every 𝛾𝛾 > 0 and for each 𝑦𝑦 ∈ 𝑌𝑌,

�(𝑚𝑚, 𝑘𝑘) ∈ ℕ × ℕ:_{𝑚𝑚𝑘𝑘 �}1 𝑚𝑚𝑘𝑘 𝑖𝑖,𝑗𝑗=1,1

𝑓𝑓(|𝑑𝑑(𝑦𝑦; 𝐶𝐶𝑖𝑖𝑗𝑗, 𝐷𝐷𝑖𝑖𝑗𝑗) − 𝐿𝐿|) ≥ 𝛾𝛾� ∈ ℐ2𝜎𝜎

then, {𝐶𝐶_{𝑖𝑖𝑗𝑗}} and {𝐷𝐷_{𝑖𝑖𝑗𝑗}} are said to be strongly 𝑓𝑓 -asymptotically ℐ₂𝜎𝜎-equivalent of multiple 𝐿𝐿 (denoted by 𝐶𝐶𝑖𝑖𝑗𝑗 ~

[𝑊𝑊_ℐ2𝜎𝜎𝐿𝐿(𝑓𝑓)]

𝐷𝐷𝑖𝑖𝑗𝑗) and if 𝐿𝐿 = 1, then {𝐶𝐶𝑖𝑖𝑗𝑗} and {𝐷𝐷𝑖𝑖𝑗𝑗} are said to be strongly 𝑓𝑓-asymptotically ℐ2𝜎𝜎-equivalent.

A double sequence 𝜃𝜃₂= {(𝑘𝑘_𝑟𝑟, 𝑗𝑗_𝑢𝑢)} is called double lacunary sequence if there exist two increasing sequence of integers such that

𝑘𝑘0= 0, ℎ𝑟𝑟= 𝑘𝑘𝑟𝑟− 𝑘𝑘𝑟𝑟−1→ ∞ 𝑎𝑎𝑛𝑛𝑑𝑑 𝑗𝑗0= 0, ℎ�𝑢𝑢= 𝑗𝑗𝑢𝑢− 𝑗𝑗𝑢𝑢−1→ ∞, 𝑎𝑎𝑒𝑒 𝑟𝑟, 𝑢𝑢 → ∞. We use the following notations afterwards:

𝑘𝑘𝑟𝑟𝑢𝑢= 𝑘𝑘𝑟𝑟𝑗𝑗𝑢𝑢, ℎ𝑟𝑟𝑢𝑢= ℎ𝑟𝑟ℎ�𝑢𝑢, 𝐼𝐼𝑟𝑟𝑢𝑢 = {(𝑘𝑘, 𝑗𝑗): 𝑘𝑘𝑟𝑟−1< 𝑘𝑘 ≤ 𝑘𝑘𝑟𝑟 𝑎𝑎𝑛𝑛𝑑𝑑 𝑗𝑗𝑢𝑢−1< 𝑗𝑗 ≤ 𝑗𝑗𝑢𝑢}.

After this, we take 𝜃𝜃₂= {(𝑘𝑘_𝑟𝑟, 𝑗𝑗_𝑢𝑢)} as a double lacunary sequence. Let 𝜃𝜃₂= {(𝑘𝑘_𝑟𝑟, 𝑗𝑗_𝑢𝑢)} be a double lacunary sequence, 𝐶𝐶 ⊆ ℕ × ℕ and

𝑒𝑒𝑟𝑟𝑢𝑢≔ min_{𝑚𝑚,𝑛𝑛} �𝐶𝐶 ∩ ��𝜎𝜎𝑘𝑘(𝑚𝑚), 𝜎𝜎𝑗𝑗(𝑛𝑛)� : (𝑘𝑘, 𝑗𝑗) ∈ 𝐼𝐼𝑟𝑟𝑢𝑢�� and 𝑆𝑆𝑟𝑟𝑢𝑢≔ max 𝑚𝑚,𝑛𝑛 �𝐶𝐶 ∩ ��𝜎𝜎 𝑘𝑘_{(𝑚𝑚), 𝜎𝜎}𝑗𝑗_{(𝑛𝑛)� : (𝑘𝑘, 𝑗𝑗) ∈ 𝐼𝐼} 𝑟𝑟𝑢𝑢��.

If the limits 𝑉𝑉₂𝜃𝜃(𝐶𝐶): = lim 𝑟𝑟,𝑢𝑢→∞ 𝑠𝑠𝑟𝑟𝑟𝑟 ℎ𝑟𝑟𝑟𝑟 and 𝑉𝑉2 𝜃𝜃_{(𝐶𝐶): = lim} 𝑟𝑟,𝑢𝑢→∞ 𝑆𝑆𝑟𝑟𝑟𝑟

ℎ𝑟𝑟𝑟𝑟 exist, then they are called a lower lacunary 𝜎𝜎-uniform

density and an upper lacunary 𝜎𝜎-uniform density of the set 𝐶𝐶, respectively. If 𝑉𝑉₂𝜃𝜃(𝐶𝐶) = 𝑉𝑉₂𝜃𝜃(𝐶𝐶), then 𝑉𝑉₂𝜃𝜃(𝐶𝐶) = 𝑉𝑉2𝜃𝜃(𝐶𝐶) = 𝑉𝑉2𝜃𝜃(𝐶𝐶) is called the lacunary 𝜎𝜎-uniform density of 𝐶𝐶.

After this, we take ℐ₂𝜎𝜎𝜃𝜃 as a strongly admissible ideal in ℕ × ℕ.

Lemma 1 [33] Let f be a modulus and 0 < 𝛿𝛿 < 1. Then, for each u ≥ γ we have f(u) ≤ 2f(1)γ−1u.

Method

In the proofs of the theorems obtained in this study, used frequently in mathematics,

i.Direct proof method,

ii. Reverse proof method,

iii.Contrapositive method, iv. Induction method

methods were used as needed.

Definition 2.1. If for every 𝛾𝛾 > 0 and each 𝑦𝑦 ∈ 𝑌𝑌,

�(𝑟𝑟, 𝑢𝑢) ∈ ℕ × ℕ:_ℎ1 𝑟𝑟𝑢𝑢_{(𝑘𝑘,𝑗𝑗)∈𝐼𝐼}�

𝑟𝑟𝑟𝑟

|𝑑𝑑(𝑦𝑦; 𝐶𝐶𝑘𝑘𝑗𝑗, 𝐷𝐷𝑘𝑘𝑗𝑗) − 𝐿𝐿| ≥ 𝛾𝛾� ∈ ℐ2𝜎𝜎𝜃𝜃,

then the double sequences {𝐶𝐶_{𝑘𝑘𝑗𝑗}} and {𝐷𝐷_{𝑘𝑘𝑗𝑗}} are said to be strongly asymptotically ℐ₂𝜎𝜎𝜃𝜃-equivalent of multiple 𝐿𝐿 denoted by

𝐶𝐶𝑘𝑘𝑗𝑗 ~ [𝑊𝑊

ℐ2𝜎𝜎𝜎𝜎𝐿𝐿 ]

𝐷𝐷𝑘𝑘𝑗𝑗

and if 𝐿𝐿 = 1, then {𝐶𝐶_{𝑘𝑘𝑗𝑗}} and {𝐷𝐷_{𝑘𝑘𝑗𝑗}} are said to be strongly asymptotically ℐ₂𝜎𝜎𝜃𝜃-equivalent.

Definition 2.2. If for every 𝛾𝛾 > 0 and each 𝑦𝑦 ∈ 𝑌𝑌,

�(𝑘𝑘, 𝑗𝑗) ∈ ℕ × ℕ: 𝑓𝑓(|𝑑𝑑(𝑦𝑦; 𝐶𝐶𝑘𝑘𝑗𝑗, 𝐷𝐷𝑘𝑘𝑗𝑗) − 𝐿𝐿|) ≥ 𝛾𝛾� ∈ ℐ2𝜎𝜎𝜃𝜃,

then the double sequences {𝐶𝐶_{𝑘𝑘𝑗𝑗}} and {𝐷𝐷_{𝑘𝑘𝑗𝑗}} are said to be 𝑓𝑓-asymptotically ℐ₂𝜎𝜎𝜃𝜃-equivalent of multiple 𝐿𝐿 denoted by 𝐶𝐶_{𝑘𝑘𝑗𝑗} ~

𝑊𝑊

ℐ2𝜎𝜎𝜎𝜎𝐿𝐿 (𝑓𝑓)

𝐷𝐷𝑘𝑘𝑗𝑗

and if 𝐿𝐿 = 1, then {𝐶𝐶_{𝑘𝑘𝑗𝑗}} and {𝐷𝐷_{𝑘𝑘𝑗𝑗}} are said to be 𝑓𝑓-asymptotically ℐ₂𝜎𝜎𝜃𝜃-equivalent.

Definition 2.3 If for every 𝛾𝛾 > 0 and each 𝑦𝑦 ∈ 𝑌𝑌,

�(𝑟𝑟, 𝑢𝑢) ∈ ℕ × ℕ:_ℎ1 𝑟𝑟𝑢𝑢_{(𝑘𝑘,𝑗𝑗)∈𝐼𝐼}�

𝑟𝑟𝑟𝑟

𝑓𝑓(|𝑑𝑑(𝑦𝑦; 𝐶𝐶𝑘𝑘𝑗𝑗, 𝐷𝐷𝑘𝑘𝑗𝑗) − 𝐿𝐿|) ≥ 𝛾𝛾� ∈ ℐ2𝜎𝜎𝜃𝜃,

then the double sequences {𝐶𝐶_{𝑘𝑘𝑗𝑗}} and {𝐷𝐷_{𝑘𝑘𝑗𝑗}} are said to be strongly 𝑓𝑓-asymptotically ℐ₂𝜎𝜎𝜃𝜃-equivalent of multiple 𝐿𝐿 denoted by

𝐶𝐶𝑘𝑘𝑗𝑗 ~ [𝑊𝑊

ℐ2𝜎𝜎𝜎𝜎𝐿𝐿 (𝑓𝑓)]

𝐷𝐷𝑘𝑘𝑗𝑗

and if 𝐿𝐿 = 1, then {𝐶𝐶_{𝑘𝑘𝑗𝑗}} and {𝐷𝐷_{𝑘𝑘𝑗𝑗}} are said to be strongly 𝑓𝑓-asymptotically ℐ₂𝜎𝜎𝜃𝜃-equivalent.

Theorem 2.1. For each 𝑦𝑦 ∈ 𝑌𝑌, we have

𝐶𝐶𝑘𝑘𝑗𝑗 ~ [𝑊𝑊 ℐ2𝜎𝜎𝜎𝜎 𝐿𝐿 _] 𝐷𝐷𝑘𝑘𝑗𝑗⇒ 𝐶𝐶𝑘𝑘𝑗𝑗 ~ [𝑊𝑊 ℐ2𝜎𝜎𝜎𝜎 𝐿𝐿 _(𝑓𝑓)] 𝐷𝐷𝑘𝑘𝑗𝑗. Theorem 2.2. If 𝑙𝑙𝑒𝑒𝑚𝑚 𝑧𝑧→∞ 𝑓𝑓(𝑧𝑧) 𝑧𝑧 = 𝛼𝛼 > 0 , then 𝐶𝐶𝑘𝑘𝑗𝑗 ~ [𝑊𝑊 ℐ2𝜎𝜎𝜎𝜎𝐿𝐿 ] 𝐷𝐷𝑘𝑘𝑗𝑗 ⇔ 𝐶𝐶𝑘𝑘𝑗𝑗 ~ [𝑊𝑊 ℐ2𝜎𝜎𝜎𝜎𝐿𝐿 (𝑓𝑓)] 𝐷𝐷𝑘𝑘𝑗𝑗. References

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