pith. sign in

arxiv: 2506.13103 · v4 · submitted 2025-06-16 · 🧮 math.CA · math.HO

Descriptions of Cantor Sets: A Set-Theoretic Survey and Open Problems

Mohsen Soltanifar This is my paper

Pith reviewed 2026-05-19 09:57 UTC · model grok-4.3

classification 🧮 math.CA math.HO
keywords Cantor setsdescriptive set theoryBorel hierarchyiterated function systemsmeasure zeropositive measuremiddle-third Cantor setq-ary expansions
0
0 comments X

The pith

The classical middle-third Cantor set belongs to three distinct families of Cantor sets on the real line, spanning both measure-zero and positive-measure examples.

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

This survey compiles historical and set-theoretic descriptions of deterministic Cantor sets and arranges them into four representations ordered from most general to most specific. It supplies explicit and recursive constructions for two families of thin measure-zero Cantor sets and one augmented tick family of positive measure. The central observation is that the middle-third Cantor set sits inside all three families at once. A reader would care because the result unifies classical examples with broader descriptive classifications and flags open directions for extending the taxonomy.

Core claim

The paper reviews the Borel hierarchy and then organizes deterministic Cantor sets via four hierarchically ordered representations: general, nested, iterated-function-system (IFS), and q-ary expansion. It gives concrete constructions for two thin families of measure-zero Cantor sets and an augmented tick family of positive measure. The classical middle-third Cantor set is shown to lie in the intersection of these three families.

What carries the argument

The four hierarchically ordered representations (general, nested, IFS, and q-ary expansion) that organize set-theoretic descriptions of deterministic Cantor sets from most general to most specific.

If this is right

  • The middle-third set admits descriptions from each of the three families simultaneously.
  • The taxonomy supplies a common platform for comparing other Cantor-type sets against the same three families.
  • Open problems isolated in four directions become natural next steps for extending the hierarchy and constructions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The hierarchy could be tested against fat Cantor sets or other positive-measure examples to see whether they also fall into the tick family.
  • Connections between the q-ary expansions and the Borel hierarchy might yield finer classification of the descriptive complexity of these sets.
  • Recursive descriptions from the tick family could be adapted to construct new examples with prescribed Hausdorff dimension.
  • The survey's organizational framework may generalize to Cantor sets in higher dimensions or in other metric spaces.

Load-bearing premise

The four representations form a strict hierarchy from most general to most specific set-theoretic description of deterministic Cantor sets.

What would settle it

A deterministic Cantor set on the real line that satisfies the middle-third construction yet cannot be expressed inside at least one of the three families, or a member of the tick family whose Lebesgue measure is shown to be zero.

Figures

Figures reproduced from arXiv: 2506.13103 by Mohsen Soltanifar.

Figure 1
Figure 1. Figure 1: Diagram of descriptive formulas for the Cantor sets from set-theoretic perspective: C refers to the middle-third [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

This survey synthesizes the principal descriptive set-theoretic perspectives on deterministic Cantor sets on the real line and charts directions for future study. After recounting their historical genesis and compiling an up-to-date taxonomy, we review the Borel hierarchy and four hierarchically ordered representations-general, nested, iterated-function-system (IFS), and q-ary expansion-presented from the most general to the most specific set-theoretic description of deterministic Cantor sets. We then present explicit and recursive descriptions for two thin families of measure-zero Cantor sets and an augmented "tick" family of positive measure, respectively, showing that the classical middle-third set lies in the intersection of all three families of after-mentioned Cantor sets. The survey closes by isolating several open problems in four directions, aiming to provide mathematicians with a coherent platform for further descriptive set-theoretic investigations into Cantor-type sets on the real line.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript is a survey synthesizing descriptive set-theoretic perspectives on deterministic Cantor sets on the real line. It recounts their historical genesis, compiles an up-to-date taxonomy, reviews the Borel hierarchy, and presents four hierarchically ordered representations (general, nested, iterated-function-system (IFS), and q-ary expansion) from most general to most specific. Explicit and recursive descriptions are provided for two thin families of measure-zero Cantor sets and an augmented 'tick' family of positive measure, with the claim that the classical middle-third Cantor set lies in the intersection of all three families. The survey concludes by isolating open problems in four directions.

Significance. If the explicit and recursive descriptions hold and the membership of the middle-third set in the three families is correctly established, the paper supplies a coherent organizational framework and concrete constructions that could serve as a useful reference for further work in descriptive set theory and real analysis. The hierarchical ordering of representations and the list of open problems provide a platform for targeted investigations into Cantor-type sets.

major comments (2)
  1. The abstract and the section presenting the families state that the augmented tick family consists of positive-measure sets, yet claim that the middle-third Cantor set (which has Lebesgue measure zero) belongs to this family and thus to the intersection of all three families. The manuscript must clarify whether the augmentation explicitly permits zero-measure members or whether 'positive measure' is not a strict membership requirement for every set in the family; without this, the intersection claim rests on an unresolved definitional point.
  2. In the organizational framework for the four representations, the assertion that they form a strict hierarchy (general to nested to IFS to q-ary) requires explicit inclusion relations or counterexamples demonstrating that each level is a proper specialization of the previous. If the hierarchy is only illustrative rather than rigorously inclusion-based, the claim that the q-ary expansion is the most specific description needs adjustment.
minor comments (2)
  1. Verify that all historical references and recent citations in the taxonomy section are complete and correctly attributed.
  2. Ensure consistent use of notation when distinguishing the thin families from the augmented tick family across the explicit and recursive descriptions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and insightful review of our manuscript. We address each of the major comments in detail below and outline the revisions we intend to make.

read point-by-point responses

  1. Referee: The abstract and the section presenting the families state that the augmented tick family consists of positive-measure sets, yet claim that the middle-third Cantor set (which has Lebesgue measure zero) belongs to this family and thus to the intersection of all three families. The manuscript must clarify whether the augmentation explicitly permits zero-measure members or whether 'positive measure' is not a strict membership requirement for every set in the family; without this, the intersection claim rests on an unresolved definitional point.

    Authors: We appreciate the referee's identification of this potential inconsistency in our description. In constructing the augmented 'tick' family, the primary members are designed to have positive Lebesgue measure by incorporating additional intervals ('ticks') that contribute positive measure. The classical middle-third Cantor set is incorporated into this family through a specific choice of parameters in the augmentation that effectively reduces the measure to zero. To resolve the definitional ambiguity, we will revise both the abstract and the section on the families to clarify that the augmented tick family generally consists of positive-measure Cantor sets, but the augmentation explicitly allows for zero-measure instances, including the middle-third set as a distinguished member. This revision will be implemented in the next version of the manuscript. revision: yes

  2. Referee: In the organizational framework for the four representations, the assertion that they form a strict hierarchy (general to nested to IFS to q-ary) requires explicit inclusion relations or counterexamples demonstrating that each level is a proper specialization of the previous. If the hierarchy is only illustrative rather than rigorously inclusion-based, the claim that the q-ary expansion is the most specific description needs adjustment.

    Authors: We agree that making the hierarchical structure rigorous will enhance the clarity of our organizational framework. In the revised manuscript, we will explicitly establish the inclusion relations: every nested representation is a special case of a general representation, every IFS representation is a nested one, and every q-ary expansion is an IFS representation. Furthermore, we will supply counterexamples to demonstrate that these inclusions are proper. For instance, we will exhibit a general Cantor set that cannot be represented as nested, a nested set that is not an IFS, and an IFS that does not admit a q-ary expansion in the required sense. This will confirm the strict hierarchy and justify the claim that the q-ary expansion provides the most specific description. We will add this material to the section discussing the four representations. revision: yes

Circularity Check

0 steps flagged

No circularity: survey taxonomy and hierarchy are classificatory, not derived from self-referential inputs

full rationale

This is a descriptive survey paper that compiles historical and set-theoretic perspectives on Cantor sets, presents a taxonomy of representations (general, nested, IFS, q-ary), and defines three families of Cantor sets with explicit or recursive descriptions. The central claim that the middle-third Cantor set belongs to the intersection of these families follows directly from checking membership against the provided definitions of the families, without any fitted parameters, self-citations that bear the load of a uniqueness theorem, or renamings that substitute for derivation. The organizational hierarchy is stated as a framework for presentation rather than a theorem derived from prior results within the paper. No load-bearing step reduces to an input by construction; the work is self-contained against external benchmarks in the cited literature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a survey paper the work does not introduce new free parameters, axioms, or invented entities; it reviews and organizes existing set-theoretic descriptions and representations from the literature.

pith-pipeline@v0.9.0 · 5667 in / 1056 out tokens · 21102 ms · 2026-05-19T09:57:11.423062+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

60 extracted references · 60 canonical work pages · 11 internal anchors

  1. [1]

    Fleron, J. F. (1994). A note on the history of the cantor set and cantor function. Mathematics Magazine, 67(2), 136–140.https://doi.org/10.1080/0025570x.1994.11996201

    work page doi:10.1080/0025570x.1994.11996201 1994

  2. [2]

    Edgar, G. (2007). Measure, topology, and fractal geometry.Springer, New York, NY , USA

    work page 2007

  3. [3]

    Vallin, R. W. (2013). The elements of cantor sets: With Applications. John Wiley & Sons, Hoboken, NJ, USA

    work page 2013

  4. [4]

    Falconer, K. (2014). Fractal geometry: Mathematical Foundations and Applications . John Wiley & Sons, Chichester, West Sussex, UK

    work page 2014

  5. [5]

    Gorodetski, A., & Northrup, S. (2015). On Sums of Nearly Affine Cantor Sets.arxiv - Cornell university. Pre-print. https://doi.org/10.48550/arxiv.1510.07008

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1510.07008 2015

  6. [6]

    Lin, Z., Qiu, Y ., & Wang, K. (2024). Number rigid determinantal point processes induced by generalized Cantor sets. arxiv - Cornell university. Pre-print.https://doi.org/10.48550/arxiv.2407.14168

    work page doi:10.48550/arxiv.2407.14168 2024

  7. [7]

    McLoughlin, M. P. M. M., & Krizan, C. R. (2014). Some results on distorted Cantor sets. International Journal of Mathematical Analysis, 8, 35–49.https://doi.org/10.12988/ijma.2014.311279

    work page doi:10.12988/ijma.2014.311279 2014

  8. [8]

    Jumha, Sebar H.; Hamad, Ibrahim O.; and Hassan, Ala O. (2024). Cantor Set from a Nonstandard Viewpoint, Polytechnic Journal, 14(1), Article 12.https://doi.org/10.59341/2707-7799.1830

    work page doi:10.59341/2707-7799.1830 2024

  9. [9]

    Nassiri, M., & Bidaki, M. Z. (2025). Cantor sets in higher dimension I: Criterion for stable intersections. arxiv - Cornell university. Pre-print.https://arxiv.org/abs/2502.02906

    work page arXiv 2025

  10. [10]

    Nowakowski, P. (2021). The Family of Central Cantor Sets with Packing Dimension Zero. Tatra Mountains Mathematical Publications, 78(1), 1–8.https://doi.org/10.2478/tmmp-2021-0001

    work page doi:10.2478/tmmp-2021-0001 2021

  11. [11]

    Batakis, A., & Zdunik, A. (2015). Hausdorff and harmonic measures on non-homogeneous Cantor sets. Annales Academiae Scientiarum Fennicae Mathematica, 40, 279–303.https://doi.org/10.5186/aasfm.2015.4012

    work page doi:10.5186/aasfm.2015.4012 2015

  12. [12]

    Batakis, A., & Havard, G. (2024). Dimensions of harmonic measures on non-autonomous Cantor sets. arxiv - Cornell university. Pre-print.https://arxiv.org/abs/2409.08019

    work page arXiv 2024

  13. [13]

    Bellissard, J., V ., & Julien, A. (2013). Bi-Lipshitz Embedding of Ultrametric Cantor Sets into Euclidean Spaces. arxiv - Cornell university. Pre-print.https://arxiv.org/abs/1202.4330

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  14. [14]

    Hieronymi, P. (2018). A tame Cantor set. Journal of the European Mathematical Society , 20(9), 2063–2104. https://doi.org/10.4171/jems/806 11 arXiv Template A Preprint

    work page doi:10.4171/jems/806 2018

  15. [15]

    Hu, M., & Wen, S. (2008). Quasisymmetrically minimal uniform Cantor sets. Topology and Its Applications, 155(6), 515–521.https://doi.org/10.1016/j.topol.2007.10.006

    work page doi:10.1016/j.topol.2007.10.006 2008

  16. [16]

    Extensive analytical and numerical investigation of the kinetic and stochastic Cantor set

    Hassan, M. K., Hassan, M. Z., & Pavel, N. I. (2018). Extensive analytical and numerical investigation of the kinetic and stochastic Cantor set. arxiv - Cornell university Pre-print:0907.4837

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  17. [17]

    Garcia, I. (2011). A family of smooth Cantor sets. Annales Fennici Mathematici, 36(1), 21-45

    work page 2011

  18. [18]

    Fillman, J., & Tidwell, S. H. (2023). On Sums of Semibounded Cantor Sets. Rocky Mountain Journal of Mathematics, 53(3), 737-754

    work page 2023

  19. [19]

    Denette, E. (2016). Minimal cantor sets: The combinatorial construction of ergodic families and semi-conjugations. Open Access Dissertations. Paper 436.https://digitalcommons.uri.edu/oa_diss/436

    work page 2016

  20. [20]

    Tresser, C. (1991). Fine Structure of Universal Cantor Sets. In: Tirapegui, E., Zeller, W. (eds)Instabilities and Nonequilibrium Structures III. Mathematics and Its Applications, vol 64. Springer, Dordrecht, Netherlands.https: //doi.org/10.1007/978-94-011-3442-2_3

    work page doi:10.1007/978-94-011-3442-2_3 1991

  21. [21]

    G., Pacifico, M., & Romaña Ibarra, S

    Moreira, C. G., Pacifico, M., & Romaña Ibarra, S. (2020). Hausdorff dimension, Lagrange and Markov dynamical spectra for geometric Lorenz attractors. Bulletin of the American Mathematical Society, 57(2), 269-292

    work page 2020

  22. [22]

    L., & Lu , H

    Lapidus, M. L., & Lu , H. (2008). Nonarchimedean Cantor set and string. Journal of Fixed Point Theory and Applications, 3(1), 181-190

    work page 2008

  23. [23]

    Cheraghi, D., & Pedramfar, M. (2019). Hairy Cantor sets. arxiv - Cornell university. Pre-print, https://doi. org/10.48550/arxiv.1907.03349

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1907.03349 2019

  24. [24]

    Cabrelli, C., Darji, U., & Molter, U. (2011). Visible and Invisible Cantor sets. arxiv - Cornell university Pre-print. https://doi.org/10.48550/arxiv.1109.1174

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1109.1174 2011

  25. [25]

    Cockerill, E. (2002). Multi-Model Cantor Sets. arxiv - Cornell university . Pre-print. https://doi.org/10. 48550/arxiv.math/0202198

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  26. [26]

    Hare, K., & Ng, K.-S. (2015). Hausdorff and Packing Measures of Balanced Cantor Sets. Real Analysis Exchange, 40(1), 113.https://doi.org/10.14321/realanalexch.40.1.0113

    work page doi:10.14321/realanalexch.40.1.0113 2015

  27. [27]

    Zeng, Y . (2017). SELF-SIMILAR SUBSETS OF SYMMETRIC CANTOR SETS.Fractals, 25(01), 1750003. https://doi.org/10.1142/s0218348x17500037

    work page doi:10.1142/s0218348x17500037 2017

  28. [28]

    Zhou, Q., Li, X., & Li, Y . (2019). Deformations on symbolic Cantor sets and ultrametric spaces.arxiv - Cornell university. Pre-print.https://doi.org/10.48550/arxiv.1911.01017

    work page doi:10.48550/arxiv.1911.01017 2019

  29. [29]

    Honary, B., Pourbarat, M., & Velayati, M. R. (2006). On the Arithmetic Difference of Affine Cantor Sets.Journal of Dynamical Systems and Geometric Theories, 4(2), 139–146.https://doi.org/10.1080/1726037x.2006. 10698511

    work page doi:10.1080/1726037x.2006 2006

  30. [30]

    Fletcher, A., & Wu, J. (2014). Julia Sets and wild Cantor sets. arxiv - Cornell university . Pre-print. https: //doi.org/10.48550/arxiv.1403.6790

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1403.6790 2014

  31. [31]

    Akhadkulov, H., & Jiang, Y . (2019). V olume of the hyperbolic cantor sets.Dynamical Systems, 35(2), 185–196. https://doi.org/10.1080/14689367.2019.1659754

    work page doi:10.1080/14689367.2019.1659754 2019

  32. [32]

    Babich, A. (1992). Scrawny Cantor sets are not definable by tori. Proceedings of the American Mathematical Society, 115(3), 829.https://doi.org/10.1090/s0002-9939-1992-1106178-4

    work page doi:10.1090/s0002-9939-1992-1106178-4 1992

  33. [33]

    Krushkal, V . (2018). Sticky Cantor sets in Rd. Journal of Topology and Analysis , 10(02), 477–482. https: //doi.org/10.1142/s1793525318500164

    work page doi:10.1142/s1793525318500164 2018

  34. [34]

    Kamalutdinov, K., & Tetenov, A. (2018). Twofold Cantor sets in R.arxiv - Cornell university. Pre-print.https: //doi.org/10.48550/arxiv.1802.03872

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1802.03872 2018

  35. [35]

    Kong, D. (2014). On the self similarity of generalized Cantor sets. SCIENTIA SINICA Mathematica , 44(9), 945–956.https://doi.org/10.1360/n012013-00114

    work page doi:10.1360/n012013-00114 2014

  36. [36]

    Semmes, S. (2011). p-Adic Heisenberg Cantor sets, 2. arxiv - Cornell university. Pre-printhttps://doi.org/ 10.48550/arxiv.1110.0111

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1110.0111 2011

  37. [37]

    Li, W., & Yao, Y . (2008). THE POINTWISE DENSITIES OF NON-SYMMETRIC CANTOR SETS.International Journal of Mathematics, 19(09), 1121–1135.https://doi.org/10.1142/s0129167x08005035

    work page doi:10.1142/s0129167x08005035 2008

  38. [38]

    Martens, M. (1994). Distortion results and invariant Cantor sets of unimodal maps.Ergodic Theory and Dynamical Systems, 14(2), 331–349.https://doi.org/10.1017/s0143385700007902

    work page doi:10.1017/s0143385700007902 1994

  39. [39]

    Nakata, T. (1997). An approximation of Hausdorff dimensions of generalized cookie-cutter Cantor sets.Hiroshima mathematical journal, 27(3), 467-475. 12 arXiv Template A Preprint

    work page 1997

  40. [40]

    Rani, M., & Prasad, S. K. (2010). Superior Cantor Sets and Superior Devil Staircases. International Journal of Artificial Life Research, 1(1), 78–84.https://doi.org/10.4018/jalr.2010102106

    work page doi:10.4018/jalr.2010102106 2010

  41. [41]

    Fletcher, A., Stoertz, D., & Vellis, V . (2022). Genusg Cantor sets and germane Julia sets. arxiv - Cornell university Pre-print.https://doi.org/10.48550/arxiv.2210.06619

    work page doi:10.48550/arxiv.2210.06619 2022

  42. [42]

    Eswarathasan, S., & Han, X. (2023). Fractal uncertainty principle for discrete Cantor sets with random alphabets. Mathematical Research Letters, 30(6), 1657–1679.https://doi.org/10.4310/mrl.2023.v30.n6.a2

    work page doi:10.4310/mrl.2023.v30.n6.a2 2023

  43. [43]

    G., & Sidorov, N

    Hare, K. G., & Sidorov, N. (2024). The Minkowski sum of linear Cantor sets. Acta Arithmetica, 212, 173-193

    work page 2024

  44. [44]

    Bartoszewicz, A., Filipczak, M., Głcab, S., Prus-Wi´sniowski, F., & Swaczyna, J. (2017). On generating regular Cantorvals connected with geometric Cantor sets. arxiv - Cornell university . Pre-print.https://doi.org/10. 48550/arxiv.1706.03523

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  45. [45]

    Mantica, G. (2015). Numerical computation of the isospectral torus of finite gap sets and of IFS Cantor sets. arxiv - Cornell university. Pre-print.https://doi.org/10.48550/arxiv.1503.03801

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1503.03801 2015

  46. [46]

    Soltanifar, M. (2023). A Classification of Elements of Function Space F(R,R). Mathematics, 11(17), 3715. https://doi.org/10.3390/math11173715

    work page doi:10.3390/math11173715 2023

  47. [47]

    Soltanifar, M. (2021). A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals. Mathe- matics, 9(13), 1546.https://doi.org/10.3390/math9131546

    work page doi:10.3390/math9131546 2021

  48. [48]

    Soltanifar, M. (2022). The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals. Mathematics, 10(5), 706.https://doi.org/10.3390/math10050706

    work page doi:10.3390/math10050706 2022

  49. [49]

    Frame, M., Mandelbrot, B., & Neger, N. (2025). Department of Mathematics. Yale University. Ancient Egyptian columns [Web page]. In Fractals in African Art. Retrieved June 15, 2025, from https://gauss.math.yale. edu/fractals/Panorama/Art/AfricanArt/EgyptianColumn.html

    work page 2025

  50. [50]

    Kainth, S. P. S. (2023). A comprehensive textbook on metric spaces. Springer. Singapore

    work page 2023

  51. [51]

    JIN, Z. A. (2021). CANTOR SETS IN TOPOLOGY , ANALYSIS, AND FINANCIAL MARKETS. Department of Mathemtics. University of Chicago. Retrieved June 15, 2025, fromhttps://math.uchicago.edu/~may/ REU2021/REUPapers/Jin,Alexa.pdf

    work page 2021

  52. [52]

    Mandelbrot, B. B. (1982). The fractal geometry of nature. W. H. Freeman. New york, NY , USA

    work page 1982

  53. [53]

    Moschovakis, Y . N. (2009).Descriptive Set Theory (V ol. 155). American Mathematical Society. Santa Monica, CA, USA

    work page 2009

  54. [54]

    Kechris, A. S. (1995). Classical Descriptive Set Theory. Springer. New York, NY , USA

    work page 1995

  55. [55]

    Astels, S. (1999). Thickness measures for Cantor sets. Electronic Research Announcements of the American Mathematical Society, 5(15), 108–111.https://doi.org/10.1090/s1079-6762-99-00068-2

    work page doi:10.1090/s1079-6762-99-00068-2 1999

  56. [56]

    Lipschitz Functions

    Searcoid, Micheal.O (2006), "Lipschitz Functions", Metric Spaces, Springer undergraduate mathematics series, Springer-Verlag, New York, NY , USA. ISBN 978-1-84628-369-7

    work page 2006

  57. [57]

    Soltanifar, M. (2006). On a Sequence of Cantor fractals. Rose–Hulman Undergraduate Mathematics Journal, 7(1), 9.https://scholar.rose-hulman.edu/rhumj/vol7/iss1/9

    work page 2006

  58. [58]

    Soltanifar, M. (2006). A different description of a family of Middle-α Cantor sets. American Journal of Under- graduate Research, 5(2).https://doi.org/10.33697/ajur.2006.014

    work page doi:10.33697/ajur.2006.014 2006

  59. [59]

    Elaydi, S. (2005). An introduction to difference equations (4th ed.). Springer, New-york, NY , USA

    work page 2005

  60. [60]

    R Core Team. (2022). R: A language and environment for statistical computing (Version 4.2.0) [Computer software]. R Foundation for Statistical Computing. Vienna, Austria.https://www.R-project.org. 13

    work page 2022