Rough $I$-convergence in cone metric spaces

Amar Kumar Banerjee; Anirban Paul

REVIEW 2 major objections 5 minor 28 references

Rough $I$-convergence in cone metric spaces

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In cone metric spaces, rough I-convergence and rough I*-convergence coincide if and only if the ideal satisfies the (AP) condition, with Example 3.2 showing the condition is needed.

desk verdict The AP equivalence theorems are correct, but the one counterexample that is supposed to separate rough I- from rough I*-convergence does not actually work, and Corollary 3 is false as stated. read the letter →

arxiv 1908.02115 v1 pith:5SVZ2QJC submitted 2019-08-06 math.MG math.GN

Amar Kumar Banerjee , Anirban Paul This is my paper · ORCID

classification math.MGmath.GN MSC 40A0540A99

keywords conemetricspacesroughconvergenceI-convergenceI*-convergence(AP)conditionidealstatistical

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper extends rough convergence to cone metric spaces, where the distance between points is a vector in a real Banach space ordered by a cone rather than a real number. It defines two ideal-based notions: rough I-convergence, which requires the set of indices where the distance from the limit exceeds $r+\varepsilon$ to belong to an ideal $I$, and rough $I^*$-convergence, which requires a subsequence indexed by a set in the associated filter to be roughly convergent. The paper's central claim is that rough $I^*$-convergence implies rough $I$-convergence in general, and that if the ideal satisfies the (AP) condition the converse holds, so the rough $I$-limit set equals the rough $I^*$-limit set. It also characterizes $I$-bounded sequences by nonempty rough $I$-limit sets and, under a normal-cone assumption, translates rough $I$-convergence into ideal convergence of the distance sequence.

What carries the argument

The machinery is the pair consisting of an admissible ideal $I$ of subsets of $\mathbb{N}$ and its associated filter $F(I)$, together with the (AP) condition on $I$. Rough $I$-convergence is defined directly through membership of the 'bad index' set in $I$, while rough $I^*$-convergence is defined through a filter set $M$ on which the subsequence is roughly convergent. The (AP) condition is the mechanism that lets the proof of Theorem 3.13 pass from a countable family of filter sets $A_i$, each capturing indices where $d(x_n,x^*)$ is within $r+\ell/i$, to a single filter set $B$ on which the tail lies within $r+\varepsilon$; Lemma 3.12 supplies exactly this countable-intersection step. In the final theorems, the normality constant $K$ of a normal cone converts cone inequalities into norm inequalities, which lets the paper identify rough $I$-convergence with ideal convergence of $\{d(x_n,x^*)-r\}$ and prove the distance-sequence continuity result.

What would settle it

Run the construction of Example 3.2 with an ideal that does satisfy (AP), using a cone metric space with a non-normal cone: if a sequence is rough $I$-convergent without being rough $I^*$-convergent, then Theorem 3.13 and Corollary 3 are false. Conversely, computing the rough $I$- and $I^*$-limit sets in Example 3.2's partition ideal and finding them equal would undermine the claimed necessity of (AP).

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Extended reading notes

Core claim

The central discovery is an equivalence between two notions of rough limit under property (AP). In a cone metric space $(X,d)$, a sequence $x=\{x_n\}$ is rough $I$-convergent of degree $r$ to $x^*$ if for every $0\ll\varepsilon$ the index set $\{n: r+\varepsilon-d(x_n,x^*)\notin\operatorname{int}P\}$ lies in $I$, and rough $I^*$-convergent of degree $r$ if some $M\in F(I)$ indexes a subsequence that is roughly convergent to $x^*$. Theorem 3.11 shows $I^*$ always implies $I$. Theorem 3.13 shows that when $I$ satisfies (AP), rough $I$-convergence of degree $r$ to $x^*$ implies rough $I^*$-convergence of the same degree, so Corollary 3 identifies the rough $I$-limit set $I-\operatorname{LIM}^r_x$ with the rough $I^*$-limit set $I^*-\operatorname{LIM}^r_x$ exactly under (AP). Example 3.2 demonstrates that the (AP) hypothesis is needed: with a partition-based ideal without (AP), a sequence can be rough $I$-convergent to a whole interval of limits while failing to be rough $I^*$-convergent to one of them.

Load-bearing premise

The equivalence between rough $I$-convergence and rough $I^*$-convergence depends on the ideal satisfying the (AP) condition, and the distance-sequence theorems further assume the cone is normal with a known constant; if either hypothesis fails, the paper's arguments do not establish the stated results.

Editorial extensions

If this is right

In every cone metric space, rough $I^*$-convergence of degree $r$ implies rough $I$-convergence of degree $r$, so $I^*-\operatorname{LIM}^r_x$ is always a subset of $I-\operatorname{LIM}^r_x$.
When $I$ satisfies (AP), the two limit sets coincide, and the rough $I$-limit set is nonempty if and only if the sequence is $I$-bounded.
Rough $I$-limit sets are monotone in the roughness degree: if $r_1>r$ then $I-\operatorname{LIM}^r_x \subset I-\operatorname{LIM}^{r_1}_x$.
If a sequence is rough $I$-convergent of degree $r$, every subsequence inherits those rough $I$-limits under the same ideal.
In a cone metric space with normal cone of constant $K$, a sequence is rough $I$-convergent of degree $r$ to $x$ exactly when $\{d(x_n,x)-r\}$ is $I$-convergent to $0$ (when the shifted distances remain in $P$), and rough $I$-convergence of $x_n$ and $y_n$ at degree $r/(4K+2)$ forces $\{d(x_n,y_n)\}$ to be rough $I$-convergent to $d(x,y)$ with degree $\|r\|$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because Example 3.2 is built from the partition ideal that fails (AP), one testable extension is to check whether the same gap between $I$ and $I^*$ limits can be produced with an ideal satisfying (AP) but with a non-normal cone; if so, Theorems 3.17 and 3.18 would not survive without normality.
The translation of rough $I$-convergence into ideal convergence of the distance sequence suggests that known results for ideal convergence of real sequences could be imported directly to prove geometric facts about rough-limit sets in normal-cone metric spaces.
One implicit consequence of Corollary 3 is that the inequality $I^*-\operatorname{LIM}^r_x \subseteq I-\operatorname{LIM}^r_x$ is strict exactly in the absence of (AP), so the size of the gap between the two limit sets could serve as a quantitative measure of how far an ideal is from having (AP).

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper introduces rough I-convergence and rough I*-convergence for sequences in cone metric spaces, then proves that I* implies I, and that with (AP) I implies I*. Those definitions are new in this setting, and the main positive results are routine but correct translations of the normed-space theory. Theorems 3.5, 3.7, 3.8, 3.9, 3.11, 3.13, 3.17 and 3.18 all check out on a careful reading. They are not deep, but they are honest, standard work.

The serious problem is Example 3.2, which is meant to show that rough I-convergence does not imply rough I*-convergence when the ideal lacks (AP). The example is invalid. With the ideal and sequence as defined there, for any fixed positive roughness degree r, take x* = r*/2 and M = union_{j>=q} D_j where 1/q <= (3/2)r*. Then M lies in F(I), and on M the sequence is within r of x*. So the sequence is actually rough I*-convergent of degree r to x*, contradicting the example. The authors choose the roughness degree after the index p that depends on the supposed M, rather than keeping r fixed. This is not a minor typo: it removes the only evidence for the claimed separation without (AP). Corollary 3's "only if" direction is also false as a general statement, since a constant sequence has equal limit sets for any ideal.

There is a smaller caveat: Theorem 3.18 and related results assume a normal cone with normal constant K, and the proofs genuinely use that assumption. That is fine as an explicit hypothesis, but it should be flagged more clearly.

The citation pattern is acceptable. The paper leans on the authors' own earlier work [7] and on standard sources [9,12,22], but the new claims are not circular and the self-citations are legitimate background.

Who is this for? Specialists in summability theory and cone metric fixed point work. It is a niche extension, not a major breakthrough, and the main proof idea is borrowed from earlier rough convergence literature. Still, it deserves a serious referee rather than a desk reject, because the definitions are reasonable and the positive theorems are sound. A referee should ask for a corrected example or a restated corollary. I would not cite it in my own work, but it is citable once the example is fixed.

Referee Report

2 major / 5 minor

Summary. The paper extends the theory of rough I-convergence to cone metric spaces. It defines rough I-convergence and rough I*-convergence of sequences of roughness degree r, studies basic properties such as I-boundedness, limit sets, cluster points, and subsequences, and proves that rough I*-convergence implies rough I-convergence (Theorem 3.11) and that under the (AP) condition on the ideal rough I-convergence implies rough I*-convergence (Theorem 3.13). It then claims, via Example 3.2, that without (AP) the two notions differ, and concludes in Corollary 3 that the rough I-limit set equals the rough I*-limit set if and only if the ideal satisfies (AP). Additional results under a normality assumption on the cone (Theorems 3.17 and 3.18) relate rough I-convergence to I-convergence of the distance sequence in the Banach space.

Significance. If the main claims were correct, the paper would provide a useful extension of known rough convergence and ideal convergence results to cone metric spaces, and the (AP)-based equivalence would clarify the relationship between rough I- and rough I*-convergence in a general setting. The paper also has genuine strengths: Theorems 3.11 and 3.13 appear correct, the use of the standard filter/ideal machinery is appropriate, and several of the auxiliary results (e.g., Theorems 3.2, 3.5, 3.7, 3.9) are routine but correct. However, the central advertised separation of rough I-convergence from rough I*-convergence without (AP) rests on Example 3.2, and that example is false. Moreover, Corollary 3's 'only if' assertion is false as stated. The paper therefore does not establish its main conceptual contribution, and the correctness of the remaining positive results is not enough to compensate for this.

major comments (2)

[Example 3.2, Section 3] The counterexample is invalid, and in fact the sequence defined there is rough I*-convergent to x* = r*/2 for every r = (r1,r2) with r* = min(r1,r2) > 0. Choose q ∈ N such that 1/q ≤ (3/2)r*, and set M = ⋃_{j≥q} D_j. Since N∖M = D_1 ∪ ⋯ ∪ D_{q-1} belongs to I, we have M ∈ F(I). For every m ∈ D_j ⊆ M with j ≥ q, x_m = 1/j and |x_m − x*| ≤ r*, so d(x_m,x*) = (|x_m−x*|, |x_m−x*|) ≤ (r*, r*) ≤ (r1+ε1, r2+ε2) for every ε = (ε1,ε2) ∈ intP. Thus the subsequence {x_m}_{m∈M} is rough convergent of degree r to x*, meaning the full sequence is rough I*-convergent to x* by Definition 3.4. This directly contradicts the conclusion of Example 3.2. Additionally, the proof changes r after choosing M: the integer p comes from H = N∖M ∈ I, and then r is chosen so that r* = 1/(3(p+1)). That is not a valid negation of rough I*-convergence for a fixed roughness degree r.
[Corollary 3, Section 3] The 'if and only if' statement is false. The 'if' direction follows from Theorems 3.11 and 3.13, but the 'only if' direction is unsupported once Example 3.2 is discarded, and it is in fact false. Consider a constant sequence x_n = x0 in any cone metric space with any admissible ideal I, whether or not I satisfies (AP). For every fixed r with 0 << r, the rough I-limit set and the rough I*-limit set both coincide: a point y is in either set exactly when d(x0,y) << r+ε for every ε ∈ intP. For instance, the set M = N ∈ F(I) witnesses rough I*-convergence, while the defining condition for rough I-convergence is independent of the ideal because the relevant bad set is either empty or all of N. Hence equality of the two limit sets holds for constant sequences for every admissible ideal, including ideals without (AP), contradicting the claimed equivalence.

minor comments (5)

[Theorem 3.14] The proof of Theorem 3.14 chooses a set L that depends on ε and then asserts rough I*-convergence of the subsequence; to establish I*-convergence one would need a single set M ∈ F(I) working for all ε. The theorem's conclusion is nevertheless correct, because the displayed inclusion directly verifies Definition 3.1 for the subsequence.
[Theorem 3.2 proof] In the proof of Theorem 3.2, 'there exists a y ∈ x' should read 'there exists a y ∈ X'.
[Theorem 3.18 proof] In the proof of Theorem 3.18, the symbol x is used both for the limit point of {x_n} and for an arbitrary vector in intP in the definition of c; this is confusing and should be replaced by a distinct symbol such as u.
[Definition 3.2] In Definition 3.2, the expression 'M − d(, x_n)' contains a stray comma and should read 'M − d(x_n, y)'.
[Throughout] There are numerous typos and grammatical slips (e.g., 'devolopments', 'remakable', 'ﬁelds' in the introduction) that should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: rough I/I*-convergence results are derived from definitions, standard ideal-filter facts, and independent cited lemmas rather than from the conclusions they claim to prove.

full rationale

The paper's central claims (Theorem 3.11, Theorem 3.13, Corollary 3) are proved from the definitions of rough I-convergence and rough I*-convergence, using the associated filter F(I), the (AP) condition, Lemma 3.12 from Nabiev-Pehlivan-Gurdal, and Lemma 2.2 from Khani-Pourmahdian. There are no fitted parameters, no quantity called a prediction that is constructed from the data it is supposed to explain, and no uniqueness theorem imported from the authors' prior work. Self-citations to [7] supply only background definitions of rough convergence and boundedness in cone metric spaces and two elementary cone/norm lemmas used as tools in Theorems 3.17-3.18; they are not the target results of the paper and do not beg the question. A reviewer concern is that Example 3.2's claim that the sequence is not rough I*-convergent for positive r may be incorrect (a set M in F(I) with all sufficiently large D_j makes the subsequence rough-convergent to r*/2), which would invalidate the 'only if' direction of Corollary 3; however, that would be a mathematical correctness problem, not a circularity. No circular step is exhibited in the derivation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It rests on the standard ideal-convergence framework and on cone metric space assumptions, with the strongest restrictions being normality of the cone and the (AP) condition.

assumptions (5)

domain assumption The cone P is a closed, convex, pointed cone in a real Banach space, and the cone metric d takes values in P (Section 2, Definitions 2.4 and 2.6).
This is the standard setting inherited from Huang and Xian [18]; it restricts the class of spaces under study.
domain assumption The ideal I is admissible and non-trivial (Section 2).
Admissibility is needed for {n} in I and to ensure that ordinary convergence implies I-convergence; used throughout.
standard math The Archimedean property of the cone: for x in P and y in intP there is n with x << ny (Lemma 2.2, from [15]).
Used in Theorem 3.13 and Example 3.2 to find a block bound for epsilons; it holds for cones in normed spaces but is a nontrivial background fact.
domain assumption Normality of the cone P with normal constant K (assumed in Theorems 3.17 and 3.18).
A strict condition on the cone that makes the norm bounds in the proofs valid; without it the results may fail.
domain assumption The (AP) condition on the ideal in Theorem 3.13, Lemma 3.12, and Corollary 3.
A hypothesis on the ideal that is necessary for the I/I* equivalence in the proof.

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Cite this review

Pith. "Pith review of Rough $I$-convergence in cone metric spaces." pith.science (2026). https://pith.science/paper/5SVZ2QJC

@misc{pith2026190802115,
  author       = {Pith},
  title        = {Pith review of: Rough $I$-convergence in cone metric spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5SVZ2QJC}},
  note         = {Machine review of arXiv:1908.02115}
}

abstract

Here we have studied the notion of rough $I$-convergence as an extension of the idea of rough convergence in a cone metric space using ideals. We have further introduced the notion of rough $I^*$-convergence of sequences in a cone metric space to find the relationship between rough $I$ and $I^*$-convergence of sequences.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[7]

Amar Kumar Banerjee and Rahul Mondal, Rough convergence of sequences in a cone metric space , arXiv : 1805.10257v1

work page Pith review arXiv
[1]

Aytar, Rough statistical convergence, Numer

S. Aytar, Rough statistical convergence, Numer. Funct. Anal. Optim. 29 (3-4) (2008)

work page 2008
[2]

Aytar, The rough limit set and the core of a real sequence , Numer

S. Aytar, The rough limit set and the core of a real sequence , Numer. Funct. Anal. and Optimiz, 29(3-4)(2008), 291- 303

work page 2008
[3]

Balacerzak, K

M. Balacerzak, K. Dems and A. Komisarski, Statistical convergence and ideal convergence for sequences of functions , J.Math. Anal. App. 328(1)(2007), 715-729

work page 2007
[4]

V esnik 67(3) (2015) 212-221

Amar Kumar Banerjee and Apurba Banerjee, A note on I-convergence and I ∗ -convergence of sequences and nets in topological spaces, Mat. V esnik 67(3) (2015) 212-221. ROUGH I-CONVERGENCE IN CONE METRIC SPACES. 11

work page 2015
[5]

Amar Kumar Banerjee and Apurba Banerjee , I-convergence classes of sequences and nets in topological s paces, Jordan Journal of Mathematics and Statistics (JJMS) 11(1), 2018, pp 13-31

work page 2018
[6]

V esnik69, 2(2017), 144-152, June 2017

Amar Kumar Banerjee and Rahul Mondal, A note on convergence of double sequences in a topological sp ace, Mat. V esnik69, 2(2017), 144-152, June 2017

work page 2017
[8]

Amar Kumar Banerjee and Apurba Banerjee , A study on I-Cauchy sequences and I-divergence in S-metric spaces, Malaya Journal of Matematik(MJM) 6(2), 2018, pp 326-330

work page 2018

Show all 28 references

[9]

D¨ undar and C

E. D¨ undar and C. C ¸ akan,Rough I-convergence Demonstratio Mathematica, V ol. XLVII, No3 (2014)

2014
[10]

Fast, Sur la convergence statistique, Colloq

H. Fast, Sur la convergence statistique, Colloq. Math 2(1951), 241-244

1951
[11]

J. A. Friday, On statistical convergence, Analysis, 5(1985), 301-313

1985
[12]

26(2), 2000/2001, pp.- 669-686

Pavel Kostyrko, Tibor ˘Sal´ at, Wladyslaw Wilczy´ nski,I-convergence, Real Analysis Exchange, V ol. 26(2), 2000/2001, pp.- 669-686

2000
[13]

Kuratowski, Topologie I, PWN W arszawa,1958

C. Kuratowski, Topologie I, PWN W arszawa,1958

1958
[14]

Kostyrko, M

P . Kostyrko, M. Macaj, T. Salat and M. Sleziak, I-convergence and extremal I-limit points, Math Slovaca, 55(2005), 443-464

2005
[15]

Khani, M

M. Khani, M. Pourmahdian, On the metrizability of cone metric spaces, Topology and its Applications, 158(2011), 190-193

2011
[16]

B. K. Lahiri and Pratulananda Das, Further results on I-limit superior and I-limit inferior , Math, Commun, 8(2003), 151-156

2003
[17]

B. K. Lahiri and Pratulananda Das, I and I ∗ -convergence in topological spaces, Math Bohemica, 130(2) (2005), 153-160

2005
[18]

Long-Guang and Z

H. Long-Guang and Z. Xian, Cone metric spaces and ﬁxed point theorems of contrapositiv e mappings, J.Math, Anal. Appl., 332 (2007), 1468- 1476

2007
[19]

Malik and M

P . Malik and M. Maity, On rough convergence of double sequence in normed linear spa ces, Bull. Allah. Math. Soc., 28(1), 89-99, 2013

2013
[20]

Malik and M

P . Malik and M. Maity, On rough statistical convergence of double sequences in nor med linear spaces, Afr . Mat., 27, 141-148, 2016

2016
[21]

Macaj , T

M. Macaj , T. Salat, Statistical convergence of subsequences of a given sequenc e, Math, Bohem, 126 (2001) 191-208

2001
[22]

Nabiev, S

A. Nabiev, S. Pehlivan, M. Gurdal, On I-Cauchy sequences, Taiwanese J.Math, 12(2) (2007), 569- 576

2007
[23]

H. X. Phu, Rough convergence in normed linear space , Numer. Funct. Anal. Optim. 22(2001), 199-222

2001
[24]

H. X. Phu, Rough convergence in inﬁnite dimensional normed spaces , Numer. Funct. Anal. Optim. 24(2003), 285-301

2003
[25]

S. K. Paul and E. Savas and H. Cakalli, I-convergence on cone metric spaces , Sarajevo Journal of Mathematics, V ol. 9(21), 85-93
[26]

˘Sal´ at,On statistically convergent sequences of real numbers , Math

T. ˘Sal´ at,On statistically convergent sequences of real numbers , Math. Slovaca 30 (1980), 139-150

1980
[27]

Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq.Math 2 (1951), 73-74

H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq.Math 2 (1951), 73-74

1951
[28]

Turkoglu and M

D. Turkoglu and M. Abuloha, Cone metric spaces and ﬁxed point theorems in diametrically contractive mappings, Acta Mathematica Sinica, English Series, March 2010, 26(3), pp 489496. (A.K.Banerjee) D EPARTMENT OF MATHEMATICS , T HE UNIVERSITY OF BURDWAN , G OLAPBAG , B URDWAN - ...

2010

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [7]

Amar Kumar Banerjee and Rahul Mondal, Rough convergence of sequences in a cone metric space , arXiv : 1805.10257v1

work page Pith review arXiv

[2] [1]

Aytar, Rough statistical convergence, Numer

S. Aytar, Rough statistical convergence, Numer. Funct. Anal. Optim. 29 (3-4) (2008)

work page 2008

[3] [2]

Aytar, The rough limit set and the core of a real sequence , Numer

S. Aytar, The rough limit set and the core of a real sequence , Numer. Funct. Anal. and Optimiz, 29(3-4)(2008), 291- 303

work page 2008

[4] [3]

Balacerzak, K

M. Balacerzak, K. Dems and A. Komisarski, Statistical convergence and ideal convergence for sequences of functions , J.Math. Anal. App. 328(1)(2007), 715-729

work page 2007

[5] [4]

V esnik 67(3) (2015) 212-221

Amar Kumar Banerjee and Apurba Banerjee, A note on I-convergence and I ∗ -convergence of sequences and nets in topological spaces, Mat. V esnik 67(3) (2015) 212-221. ROUGH I-CONVERGENCE IN CONE METRIC SPACES. 11

work page 2015

[6] [5]

Amar Kumar Banerjee and Apurba Banerjee , I-convergence classes of sequences and nets in topological s paces, Jordan Journal of Mathematics and Statistics (JJMS) 11(1), 2018, pp 13-31

work page 2018

[7] [6]

V esnik69, 2(2017), 144-152, June 2017

Amar Kumar Banerjee and Rahul Mondal, A note on convergence of double sequences in a topological sp ace, Mat. V esnik69, 2(2017), 144-152, June 2017

work page 2017

[8] [8]

Amar Kumar Banerjee and Apurba Banerjee , A study on I-Cauchy sequences and I-divergence in S-metric spaces, Malaya Journal of Matematik(MJM) 6(2), 2018, pp 326-330

work page 2018