Nonstandard Linear Combination of the Cantor Sets

Volume 28 , Issue 1 , June 2026 , Pages 115-127

Authors

Ala O. Hassan 1 ; Sebar H. Jumha 2 ; Ibrahim O. Hamad 2

1 Department of Mathematics, College of Science, Salahaddin University-Erbil, Hawler, Kurdistan Region, Iraq.

2 Department of Mathematics, College of Science, Salahaddin University-Erbil, Hawler, Kurdistan Region, Iraq

DOI logo 10.17656/sujpas.1148

Keywords

Abstract

This paper investigated the linear combinations of the Cantor set within the framework of nonstandard analysis for studying linear combinations of Cantor sets and apply it to the Minkowski sum , where  and denotes the Cantor set at an unlimited stage of its construction. The primary aim is to characterize the structure of such sums at infinitesimal resolution and to relate them rigorously to their classical counterparts. Our main result shows that  is infinitely close to the interval  in the sense of standard parts, while internally retaining a highly fragmented structure composed of infinitesimal intervals. This provides a precise nonstandard counterpart to classical results on sums of Cantor sets. The novelty of the paper lies in exploiting infinitesimal methods to capture both the microstructure and macroscopic behavior simultaneously, without resorting to limiting arguments. This yields a unified perspective on fractal sums, clarifies the role of infinitesimal gaps, and demonstrates the effectiveness of nonstandard analysis in resolving fine-scale geometric properties. The approach opens new avenues for the study of fractal sets and their applications in analysis and related fields.

References

  1. Ala, O.H., and Ibrahim, O.H., (2021): Nonstandard completion of non-complete metric space. ZANCO Journal of Pure and Applied Sciences. 33(4), pp. 129-135. http://dx.doi.org/10.21271/ZJPAS.33.4.13.
  2. Anisca, R. and Ilie, M., (2001): A technique of studying sums of central Cantor sets. Canadian Mathematical Bulletin. 44(1), pp. 12–18. https://doi.org/10.4153/CMB-2001-002-8.
  3. Bordonhos, M., Galvão, T.L.P., Gomes, J.R.B., Gouveia, J.D., Jorge, M., Lourenço, M.A.O., Pereira, J.M., Pérez-Sánchez, G., Pinto, M.L., Silva, C.M., Tedim, J., and Zêzere, B., (2023): Multiscale computational approaches toward the understanding of materials. Advanced Theory and Simulations. 6(10), 2200628. https://doi.org/10.1002/adts.202200628.
  4. Cabrélli, C.A., Hare, K.E., and Molter, U.M., (1997): Sums of Cantor sets. Ergodic Theory and Dynamical Systems. 17(6), pp. 1299–1313. https://doi.org/10.1017/S0143385797097678 .
  5. Capone, M., Romanelli, M., Castaldo, D., Parolin, G., Bello, A., Gil, G., and Vanzan, M., (2024): A vision for the future of multiscale modeling. ACS Physical Chemistry Au. 4(3), pp. 202–225. https://doi.org/10.1021/acsphyschemau.3c00080 .
  6. Chen, X., and Lv, H., (2022): New insights into the micromixer with Cantor fractal obstacles through genetic algorithm. Scientific Reports. 12, 4162. https://doi.org/10.1038/s41598-022-08144-w.
  7. Cutland, N.J., (1997): Nonstandard real analysis. In eritor name (Ed), Nonstandard Analysis: Theory and Applications. (pp. 51–76). Dordrecht. Springer Netherlands. https://doi.org/10.1007/978-94-011-5544-1_2.
  8. Diener, F. and Diener, M., (1995): Nonstandard Analysis in Practice. Berlin Heidelberg, Springer-Verlag. https://doi.org/10.1007/978-3-642-57758-1.
  9. Goldbring, I., (2014): Lecture notes on nonstandard analysis. UCLA Summer School in Logic. [online] Available at: https://www.math.uci.edu/.
  10. Gu, J., Jiang, K., Xi, L. and Zhao, B. (2021) ‘Multiplication on uniform λ-Cantor sets’, Annales Fennici Mathematici, 46, pp. 703–711. 10.5186/aasfm.2021.4643.
  11. Sebar, H.J., Ibrahim, O.H., and Ala, O.H., (2024): Cantor set from a nonstandard viewpoint. Polytechnic Journal. 14(1), pp. 104-116. https://doi.org/10.59341/2707-7799.1830.
  12. Keisler, H.J., (1976): Foundations of infinitesimal calculus. Vol. 20. Boston, Prindle, Weber & Schmidt.
  13. Kuttler, K., and Farah, I., (2025): A first course in linear algebra. Lyryx Learning Incorporated.
  14. Massopust, P.R., (2017): Fractal functions, fractal surfaces, and wavelets. 2nd ed. Amsterdam: Academic Press (Elsevier). https://doi.org/10.1016/C2015-0-01882-2.
  15. Nelson, E., (1977): Internal set theory: a new approach to nonstandard analysis. Bulletin of the American Mathematical Society. 83(6), pp. 1165–1198. https://doi.org/10.1090/s0002-9904-1977-14398-x.
  16. Parrish, R.M., Anselmetti, G.-L.R., and Gogolin, C., (2021): Analytical ground- and excited-state gradients for molecular electronic structure theory from hybrid quantum/classical methods. arXiv preprint, arXiv:2110.05040. https://doi.org/10.48550/arXiv.2110.05040.
  17. Patiño Ortiz, M., Martínez-Cruz, M.-Á., and Balankin, A.S., (2023): A brief survey of paradigmatic fractals from a topological perspective. Fractal and Fractional. 7(8), 597. https://doi.org/10.3390/fractalfract7080597.
  18. Pawłowicz, M., (2013): Linear combinations of the classic Cantor set. Tatra Mountains Mathematical Publications. 56(1), pp. 47–60. https://doi.org/10.2478/tmmp-2013-0026.
  19. Robinson, A., (1996): Non-standard analysis. 3rd ed. Princeton, NJ, Princeton University Press. https://doi.org/10.1515/9781400884223.
  20. Santos, R.A., Santos-Jr, C.V., Aguiar, E.C., Carneiro Neto, A.N., and Moura Jr, R.T., (2025): Orbital interactions in hydrogen bonds: a perspective from the chemical bond overlap model. Journal of Computational Chemistry. 46(19), e70166. https://doi.org/10.1002/jcc.70166.
  21. Soltanifar, M., (2025): Descriptions of Cantor sets: a set-theoretic survey and open problems. arXiv preprint, arXiv:2506.13103. https://doi.org/10.48550/arXiv.2506.13103.
  22. Takahashi, Y., (2019): Sums of two self-similar Cantor sets. Journal of Mathematical Analysis and Applications. 477(1), pp. 613–626. https://doi.org/10.1016/j.jmaa.2019.04.051.
  23. Wang, Z., Jiang, K., Li, W., and Zhao, B., (2021): On the sum of squares of the middle-third Cantor set. Journal of Number Theory. 218, pp. 209–222. https://doi.org/10.1016/j.jnt.2020.07.011.
  24. Wei, Y., Zhao, D., Wan, J., and Wang, D., (2022): Electron–orbital–lattice interactions in hollow multishelled structures. Trends in Chemistry. 4(11), pp. 1021–1033. https://doi.org/10.1016/j.trechm.2022.08.004.
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  • First online25 June 2026
  • Published at25 June 2026

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