Statistically characterized subgroups related to arithmetic-type sequence of integers

Ayan Ghosh; Pratulananda Das; Tamim Aziz

arxiv: 2605.16577 · v1 · pith:NMGW7YO7new · submitted 2026-05-15 · 🧮 math.GR

Statistically characterized subgroups related to arithmetic-type sequence of integers

Pratulananda Das , Ayan Ghosh , Tamim Aziz This is my paper

Pith reviewed 2026-05-19 21:02 UTC · model grok-4.3

classification 🧮 math.GR

keywords statistically characterized subgroupsarithmetic-type sequencescardinality observationsgroup theoryinteger sequencesnon-arithmetic sequencesstatistical characterization

0 comments

The pith

Statistically characterized subgroups for a broader class of arithmetic-type sequences recover all prior cardinality results as special cases but exhibit distinct behavior.

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

This paper generalizes the study of statistically characterized subgroups to a wider family of arithmetic-type sequences of integers. It shows that all earlier cardinality observations for arithmetic sequences and for certain non-arithmetic sequences arise as special cases within the new framework. The broader class, however, displays behavior that sets it apart from those narrower classes examined previously. A reader would care because the work unifies scattered earlier findings while identifying where the statistical properties of subgroups change in a substantial way.

Core claim

The authors establish that statistically characterized subgroups associated with a broader class of arithmetic-type sequences encompass all previously obtained cardinality observations for arithmetic sequences as well as certain non-arithmetic sequences as special cases. At the same time, this broader class exhibits drastically different behavior and differs significantly from the previously studied special cases.

What carries the argument

statistically characterized subgroups associated with arithmetic-type sequences

Load-bearing premise

The statistical characterization extends coherently to the broader class of arithmetic-type sequences without requiring new restrictions or producing contradictions with the special cases already studied.

What would settle it

A concrete arithmetic-type sequence from the broader class in which the cardinality of the statistically characterized subgroup does not match the pattern generalized from the special cases, or in which the overall behavior fails to differ significantly from those cases.

read the original abstract

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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper folds the authors' own recent cardinality results for statistically characterized subgroups into a single broader class of arithmetic-type sequences, recovering the old cases as special instances while flagging different behavior in the general setting.

read the letter

Hi, the main takeaway is that this manuscript extends the authors' two 2025 papers by defining a wider family of arithmetic-type sequences and showing that the earlier cardinality statements for arithmetic sequences and certain non-arithmetic ones sit inside it as special cases. They also claim the general class behaves differently from those restricted settings. That unification is the concrete advance here. If the definitions are chosen so the old results drop out cleanly without extra conditions, it does organize the existing observations under one roof and makes the contrast in behavior easier to state in one place. The abstract gives the impression that the extension is coherent and that new examples really do diverge, which is the sort of incremental cleanup that can be helpful inside a narrow research thread. On the other side, the work sits squarely on top of the same three authors' prior definitions and proofs, so anyone checking the claims will need to consult those two papers first. The phrase 'drastically different behavior' also needs the full theorems and examples to judge whether the difference is substantial or mostly quantitative. Without seeing the explicit statements of the new class and the supporting arguments, it is hard to tell how much new ground is broken versus how much is just re-packaging. This is written for people already following statistical characterizations of subgroups in group theory, especially those who have read the 2025 papers. It will not move the needle for readers outside that corner, but it could serve as a convenient reference for specialists who want the special cases collected together. I would send it for peer review. The generalization looks internally consistent on the basis of the abstract and stress-test note, and the claims are specific enough that referees can check the embedding and the claimed divergence without starting from scratch.

Referee Report

2 major / 0 minor

Summary. The manuscript studies statistically characterized subgroups of the integers for a broader class of arithmetic-type sequences. It asserts that all prior cardinality results for statistically characterized subgroups tied to arithmetic sequences and certain non-arithmetic sequences are recovered as special cases, while demonstrating that the broader class displays drastically different behavior from those special cases.

Significance. If the generalization is shown to embed the special cases coherently and to produce genuinely new phenomena without hidden restrictions, the work would unify existing cardinality observations under a single framework and identify new distinctions arising from the extended sequence class.

major comments (2)

Abstract and §1: The central claim that previous results 'follow as special cases' depends on the definitions and theorems of the two cited prior papers by the same authors; the manuscript must explicitly verify that the broader arithmetic-type class contains the earlier definitions without introducing new restrictions or altering the cardinality conclusions.
The assertion of 'drastically different behavior' requires at least one concrete theorem or example (with explicit sequence and subgroup) showing a cardinality or characterization property that fails to hold in the special cases but holds in the new class.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below and will revise the manuscript accordingly to strengthen the presentation of our claims.

read point-by-point responses

Referee: Abstract and §1: The central claim that previous results 'follow as special cases' depends on the definitions and theorems of the two cited prior papers by the same authors; the manuscript must explicitly verify that the broader arithmetic-type class contains the earlier definitions without introducing new restrictions or altering the cardinality conclusions.

Authors: We agree that an explicit verification is necessary. In the revised manuscript we will insert a short subsection immediately after the statement of our main definitions in §2. This subsection will recall the precise definitions of the arithmetic sequences and the non-arithmetic sequences appearing in the two cited papers, show that each is obtained by specializing the parameters of our broader arithmetic-type class, and confirm that the specialization introduces no additional restrictions and preserves the cardinality conclusions already established in those works. revision: yes
Referee: The assertion of 'drastically different behavior' requires at least one concrete theorem or example (with explicit sequence and subgroup) showing a cardinality or characterization property that fails to hold in the special cases but holds in the new class.

Authors: We acknowledge that a concrete illustration would make the claimed qualitative difference more transparent. While our general theorems already indicate the divergence, we will add an explicit example in the revised §4. The example will name a specific arithmetic-type sequence lying outside the classes treated in the earlier papers, identify a statistically characterized subgroup, and exhibit a cardinality (or characterization) property that is attainable in the new class but impossible for the arithmetic and previously studied non-arithmetic sequences. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper generalizes statistically characterized subgroups to a broader class of arithmetic-type sequences and states that prior cardinality results for arithmetic and certain non-arithmetic sequences are recovered as special cases while the broader class shows different behavior. Although the abstract references two recent papers by the same authors as motivation and background, the central claims involve new definitions for the extended class and independent demonstrations of differing properties. No load-bearing step reduces by construction or definition to the self-citations; the prior works supply context rather than serving as the sole justification for the new results. Self-citation is present but does not create circularity under the specified criteria, as the derivation chain for the broader class remains self-contained with external mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work operates inside standard group theory and relies on definitions of statistical characterization and arithmetic-type sequences introduced in the authors' own earlier papers; no new free parameters, ad-hoc axioms, or invented entities are visible from the abstract.

axioms (1)

standard math Standard axioms and definitions of group theory
The paper studies subgroups of groups and therefore inherits the usual group axioms.

pith-pipeline@v0.9.0 · 5655 in / 1100 out tokens · 65467 ms · 2026-05-19T21:02:53.035995+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.5. If (e_n) is weakly L-invariant then |t_s(en)(T)| = c

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

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Babieri, D

G. Babieri, D. Dikranjan, C, Milan, H. Weber, Topological torsion related to some recursive sequences of integers, Math. Nachr. 281(7) (2008), 930–950

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P. Das, A. Ghosh, Characterized subgroups related to some non-arithmetic sequence of integers, Mediterr. J. Math., 21 (2024), 164

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P. Das, A. Ghosh, Statistically characterized subgroups related to some non-arithmetic sequence of integers, Expo. Math., 43 (2025), 125653

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P. Das, A. Ghosh, T. Aziz, Topologically torsion elements of the circle group for arithmetic- type sequences, Bul. Sci. Math., 199 (2025) 103580

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P. Das, A. Ghosh, T. Aziz, Statistically characterized subgroups related to some non- arithmetic sequence of integers II (characterization for countable subgroups), J. Lond. Math. Soc., 112-6 (2025), e70365

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D. Dikranjan, Topologically torsion elements of topological groups, Topol. Proc. 26 (2001– 2002), 505–532

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D. Dikranjan, P. Das, K. Bose, Statistically characterized subgroups of the circle, Fund. Math., 249 (2020), 185-209

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Dikranjan and D

D. Dikranjan and D. Impieri, Topologically torsion elements of the circle group, Commun. Algebra, 42 (2014), 600–614

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D. Dikranjan, K. Kunen, Characterizing countable subgroups of compact abelian groups, J. Pure Appl. Algebra 208 (2007), 285–291

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D. Dikranjan, Iv. Prodanov and L. Stoyanov, Topological Groups: Characters, Dualities and Minimal Group Topologies, Pure and Applied Mathematics, Marcel Dekker Inc. New York (1989)

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Dikranjan, R

D. Dikranjan, R. Di Santo, Answer to Armacost’s quest on topologically torsion elements of the circle group, Comm. Algebra, 32 (2004), 133–146

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Dikranjan, R

D. Dikranjan, R. Di Santo, A. Giordano Bruno and H. Weber, An introduction toI- characterized subgroups of the circle, Topology Appl., 2025, to appear

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Eliaˇ s, A classification of trigonometrical thin sets and their interrelations, Proc

P. Eliaˇ s, A classification of trigonometrical thin sets and their interrelations, Proc. Amer. Math. Soc. 125(4) (1997), 1111–1121

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Fast, Sur la convergence statistique, Colloq

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A. Faisant, G. Grekos, V. Toma, On the statistical variation of sequences, J. Math. Anal. Appl., 306 (2) (2005), 432–439

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J. A. Fridy, On statistical convergence, Analysis 5(4) (1985), 301–313

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Ghosh, P

A. Ghosh, P. Das, Characterizing infinite torsion subgroups of the circle via arithmetic-type sequences, Bull. Lond. Math. Soc. 57(12) (2025), 3824–3840

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Ghosh, P

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S. Kahane, Antistable classes of thin sets in harmonic analysis, Illinois J. Math. 37 (1993), 186-223

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L. Robertson, Connectivity, divisibility, and torsion, Trans. Amer. Math. Soc. 128 (1967) 482–505

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ˇSal´ at, On statistically convergent sequences of real numbers, Mathematica Slovaca, 30(2) (1980), 139–150

T. ˇSal´ at, On statistically convergent sequences of real numbers, Mathematica Slovaca, 30(2) (1980), 139–150

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Di Santo, D

R. Di Santo, D. Dikranjan and A. Giordano Bruno, Characterized subgroups of the circle group, Ric. Mat. 67(2) (2018), 625–655

work page 2018
[38]

Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq

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

work page 1951
[39]

Weyl, ¨Uber die Gleichverteilung von Zahlen mod

H. Weyl, ¨Uber die Gleichverteilung von Zahlen mod. Eins., Math. Ann. 77(3) (1916), 313– 352

work page 1916
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Winkler, Ergodic group rotations, Hartman sets and Kronecker sequences (dedicated to Edmund Hlawka on the occasion of his 85 th birthday), Monatsh

R. Winkler, Ergodic group rotations, Hartman sets and Kronecker sequences (dedicated to Edmund Hlawka on the occasion of his 85 th birthday), Monatsh. Math. 135, No.4, 333-343 (2002)

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Zygmund, Trigonometric Series, vols

A. Zygmund, Trigonometric Series, vols. I,II, Cambridge University Press, Cambridge, New York, Melbourne (1977). Department of Mathematics, Jadavpur University, Kolkata-700032, India Email address:pratulananda@yahoo.co.in School of Mathematical and Computational Sciences, Indian Association for the Cultivation of Science, Kolkata-700032, India Email addre...

work page 1977

[1] [1]

Arbault, Sur l’ensemble de convergence absolue d’une s´ erie trigonom´ etrique., Bull

J. Arbault, Sur l’ensemble de convergence absolue d’une s´ erie trigonom´ etrique., Bull. Soc. Math. Fr. 80 (1952), 253–317

work page 1952

[2] [2]

Arhangel’skii , M

A. Arhangel’skii , M. Tkachenko, Topological Groups and Related Structures: An Introduc- tion to Topological Algebra, Atlantis Press, Paris, 2008

work page 2008

[3] [3]

Außenhofer, D

L. Außenhofer, D. Dikranjan, Locally quasi-convex compatible topologies on locally compact abelian groups, Mathematische Zeitschrift, 296 (2020), 325-351

work page 2020

[4] [4]

Barbieri, D

G. Barbieri, D. Dikranjan, A. Giordano Bruno, H. Weber, Dirichlet sets vs characterized subgroups, Topol. Appl. 231, 50–76 (2017)

work page 2017

[5] [5]

Balcerzak, K

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

work page 2007

[6] [6]

Babieri, D

G. Babieri, D. Dikranjan, C, Milan, H. Weber, Topological torsion related to some recursive sequences of integers, Math. Nachr. 281(7) (2008), 930–950

work page 2008

[7] [7]

B´ ır´ o, J.M

A. B´ ır´ o, J.M. Deshouillers, V.T. S´ os, Good approximation and characterization of subgroups ofR/Z, Studia Sci. Math. Hungar. 38 (2001), 97–113

work page 2001

[8] [8]

Borel, Sous-groupes deRli´ es ´ a la r´ epartition modulo 1 de suites, Ann

J.-P. Borel, Sous-groupes deRli´ es ´ a la r´ epartition modulo 1 de suites, Ann. Fac. Sci. Toulouse Math. 5(3–4) (1983), 217–235

work page 1983

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Borel, Sur certains sous-groupes de R li´ es ´ a la suite des factorielles, Colloq

J.-P. Borel, Sur certains sous-groupes de R li´ es ´ a la suite des factorielles, Colloq. Math. 62(1) (1991), 21–30

work page 1991

[10] [10]

R. C. Buck, The measure theoretic approach to density, Amer. J. Math. 68 (1946), 560–580

work page 1946

[11] [11]

Bukovsk´ y, N

L. Bukovsk´ y, N. Kholshchevnikova, N.N., M. Repick´ y, Thin sets in harmonic analysis and infinite combinatorics, Real Anal. Exch. 20 (1994/1995), 454–509

work page 1994

[12] [12]

Connor, V

J. Connor, V. Kadets, M. Ganichev, A Characterization of Banach Spaces with Separable Duals via Weak Statistical Convergence, J. Math. Anal. Appl., 244 (2000), 251–261

work page 2000

[13] [13]

P. Das, A. Ghosh, Solution of a general version of Armacost’s problem on topologically torsion elements, Acta Math. Hungar., 164(1) (2021), 243–264

work page 2021

[14] [14]

P. Das, A. Ghosh, On a new class of trigonometric thin sets extending Arbault sets, Bul. Sci. Math., 179 (2022), 103157

work page 2022

[15] [15]

P. Das, A. Ghosh, Characterized subgroups related to some non-arithmetic sequence of integers, Mediterr. J. Math., 21 (2024), 164

work page 2024

[16] [16]

P. Das, A. Ghosh, Statistically characterized subgroups related to some non-arithmetic sequence of integers, Expo. Math., 43 (2025), 125653

work page 2025

[17] [17]

P. Das, A. Ghosh, T. Aziz, Topologically torsion elements of the circle group for arithmetic- type sequences, Bul. Sci. Math., 199 (2025) 103580

work page 2025

[18] [18]

P. Das, A. Ghosh, T. Aziz, Statistically characterized subgroups related to some non- arithmetic sequence of integers II (characterization for countable subgroups), J. Lond. Math. Soc., 112-6 (2025), e70365

work page 2025

[19] [19]

Dikranjan, Topologically torsion elements of topological groups, Topol

D. Dikranjan, Topologically torsion elements of topological groups, Topol. Proc. 26 (2001– 2002), 505–532

work page 2001

[20] [20]

Dikranjan, P

D. Dikranjan, P. Das, K. Bose, Statistically characterized subgroups of the circle, Fund. Math., 249 (2020), 185-209

work page 2020

[21] [21]

Dikranjan and D

D. Dikranjan and D. Impieri, Topologically torsion elements of the circle group, Commun. Algebra, 42 (2014), 600–614

work page 2014

[22] [22]

Dikranjan, K

D. Dikranjan, K. Kunen, Characterizing countable subgroups of compact abelian groups, J. Pure Appl. Algebra 208 (2007), 285–291

work page 2007

[23] [23]

Dikranjan, Iv

D. Dikranjan, Iv. Prodanov and L. Stoyanov, Topological Groups: Characters, Dualities and Minimal Group Topologies, Pure and Applied Mathematics, Marcel Dekker Inc. New York (1989)

work page 1989

[24] [24]

Dikranjan, R

D. Dikranjan, R. Di Santo, Answer to Armacost’s quest on topologically torsion elements of the circle group, Comm. Algebra, 32 (2004), 133–146

work page 2004

[25] [25]

Dikranjan, R

D. Dikranjan, R. Di Santo, A. Giordano Bruno and H. Weber, An introduction toI- characterized subgroups of the circle, Topology Appl., 2025, to appear

work page 2025

[26] [26]

Di Maio, Lj.D.R

G. Di Maio, Lj.D.R. Koˇcinac, Statistical convergence in topology, Topology Appl. 156 (2008), 28–45. s-CHARACTERIZED SUBGROUPS 15

work page 2008

[27] [27]

Eliaˇ s, A classification of trigonometrical thin sets and their interrelations, Proc

P. Eliaˇ s, A classification of trigonometrical thin sets and their interrelations, Proc. Amer. Math. Soc. 125(4) (1997), 1111–1121

work page 1997

[28] [28]

Fast, Sur la convergence statistique, Colloq

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

work page 1951

[29] [29]

Faisant, G

A. Faisant, G. Grekos, V. Toma, On the statistical variation of sequences, J. Math. Anal. Appl., 306 (2) (2005), 432–439

work page 2005

[30] [30]

J. A. Fridy, On statistical convergence, Analysis 5(4) (1985), 301–313

work page 1985

[31] [31]

Ghosh, P

A. Ghosh, P. Das, Characterizing infinite torsion subgroups of the circle via arithmetic-type sequences, Bull. Lond. Math. Soc. 57(12) (2025), 3824–3840

work page 2025

[32] [32]

Ghosh, P

A. Ghosh, P. Das, Characterizing infinite torsion subgroups of the circle via arithmetic-type sequences II, submitted

work page

[33] [33]

Impieri, Characterized subgroups, PhD Thesis, July 2015

D. Impieri, Characterized subgroups, PhD Thesis, July 2015

work page 2015

[34] [34]

Kahane, Antistable classes of thin sets in harmonic analysis, Illinois J

S. Kahane, Antistable classes of thin sets in harmonic analysis, Illinois J. Math. 37 (1993), 186-223

work page 1993

[35] [35]

Robertson, Connectivity, divisibility, and torsion, Trans

L. Robertson, Connectivity, divisibility, and torsion, Trans. Amer. Math. Soc. 128 (1967) 482–505

work page 1967

[36] [36]

ˇSal´ at, On statistically convergent sequences of real numbers, Mathematica Slovaca, 30(2) (1980), 139–150

T. ˇSal´ at, On statistically convergent sequences of real numbers, Mathematica Slovaca, 30(2) (1980), 139–150

work page 1980

[37] [37]

Di Santo, D

R. Di Santo, D. Dikranjan and A. Giordano Bruno, Characterized subgroups of the circle group, Ric. Mat. 67(2) (2018), 625–655

work page 2018

[38] [38]

Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq

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

work page 1951

[39] [39]

Weyl, ¨Uber die Gleichverteilung von Zahlen mod

H. Weyl, ¨Uber die Gleichverteilung von Zahlen mod. Eins., Math. Ann. 77(3) (1916), 313– 352

work page 1916

[40] [40]

Winkler, Ergodic group rotations, Hartman sets and Kronecker sequences (dedicated to Edmund Hlawka on the occasion of his 85 th birthday), Monatsh

R. Winkler, Ergodic group rotations, Hartman sets and Kronecker sequences (dedicated to Edmund Hlawka on the occasion of his 85 th birthday), Monatsh. Math. 135, No.4, 333-343 (2002)

work page 2002

[41] [41]

Zygmund, Trigonometric Series, vols

A. Zygmund, Trigonometric Series, vols. I,II, Cambridge University Press, Cambridge, New York, Melbourne (1977). Department of Mathematics, Jadavpur University, Kolkata-700032, India Email address:pratulananda@yahoo.co.in School of Mathematical and Computational Sciences, Indian Association for the Cultivation of Science, Kolkata-700032, India Email addre...

work page 1977