e7f7f0ff-ea6a-4cb9-a3f5-2b65ec15397220230918065741884wseas:wseasmdt@crossref.orgMDT DepositPROOF2732-99412944-916210.37394/232020https://wseas.com/journals/proof/index.php22020232202023310.37394/232020.2023.3https://wseas.com/journals/proof/2023.phpMykolaYaremenkoThe National Technical University of Ukraine, Kyiv 04213, Kyiv, UKRAINEIn this article, we introduce and study a new class of perfect nowhere-dense sets, which are not selfsimilar in any subset, also, we constructed the correspondent singular functions. We construct a twodimensional irregular Cantor set Ce,π ([0, 1]) on the real plane.9152023915202329315https://wseas.com/journals/proof/2023/a10proof-003(2023).pdf10.37394/232020.2023.3.5https://wseas.com/journals/proof/2023/a10proof-003(2023).pdf10.1016/j.topol.2023.108407I. Banic, G. Erceg, S. Greenwood, J. Kennedy,“ Transitive points in CR-dynamical systems,” Topology and its Applications, V. 326, (2023). 10.1007/s11009-023-09989-yCornean, H., Herbst, I.W., Moller, J. et al., “ Singular Distribution Functions for Random Variables with Stationary Digits.” Methodol Comput Appl Probab 25, 31, (2023). 10.1016/j.heliyon.2023.e14862David R. Dellwo, “ Fat Cantor sets and their skinny companions, Heliyon,” V. 9, Issue 5, (2023). 10.1007/bf01049726K. Falconer, “ Wavelet transforms and order-two densities of fractals,” J. Statist. Phys. 67. (1992), 781- 793. 10.1017/s030500410200590xW. Li, D. Xiao, F.M. Dekking, “ Non-differentiability of devil’s staircase and dimension of subsets of Moran sets,” Math. Proc. Cambridge Philos. Soc. 133, (2002), 345–355. 10.2478/tmmp-2021-0001P. Nowakowski, “ The Family of Central Cantor Sets with Packing Dimension Zero,” Tatra Mountains Mathematical Publications, vol.78, N.1, (2021), 1-8. 10.1016/j.jmaa.2018.05.065F. Prus-Wisniowski, F. Tulone, “ The arithmetic decomposition of central Cantor sets.” J. Math. Anal. Appl. 467, (2018), 26–31. R. S. Strichartz, A. Taylor, and T. Zhang, “ Densities of self-similar measure on the line, Experiment. ” Math. 4, N. 2, (1995)