# Article

 Title: Some dimensional results for a class of special homogeneous Moran sets (English) Author: Hu, Xiaomei Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 66 Issue: 1 Year: 2016 Pages: 127-135 Summary lang: English . Category: math . Summary: We construct a class of special homogeneous Moran sets, called $\{m_{k}\}$-quasi homogeneous Cantor sets, and discuss their Hausdorff dimensions. By adjusting the value of $\{m_{k}\}_{k\ge 1}$, we constructively prove the intermediate value theorem for the homogeneous Moran set. Moreover, we obtain a sufficient condition for the Hausdorff dimension of homogeneous Moran sets to assume the minimum value, which expands earlier works. (English) Keyword: homogeneous Moran set Keyword: $\{m_{k}\}$-Moran set Keyword: $\{m_{k}\}$-quasi homogeneous Cantor set Keyword: Hausdorff dimension MSC: 28A80 idZBL: Zbl 06587879 idMR: MR3483228 DOI: 10.1007/s10587-016-0245-2 . Date available: 2016-04-07T15:00:46Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/144872 . Reference: [1] Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications.John Wiley & Sons, Chichester (1990). MR 1102677 Reference: [2] Feng, D.-J.: The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers.Adv. Math. 195 (2005), 24-101. Zbl 1078.11062, MR 2145793, 10.1016/j.aim.2004.06.011 Reference: [3] Feng, D., Wen, Z., Wu, J.: Some dimensional results for homogeneous Moran sets.Sci. China, Ser. A 40 (1997), 475-482. MR 1461002, 10.1007/BF02896955 Reference: [4] Li, J., Wu, M.: Pointwise dimensions of general Moran measures with open set condition.Sci. China, Math. 54 (2011), 699-710. Zbl 1219.28010, MR 2786709, 10.1007/s11425-011-4187-8 Reference: [5] Peng, F., Wen, S.: Fatness and thinness of uniform Cantor sets for doubling measures.Sci. China, Math. 54 (2011), 75-81. Zbl 1219.28001, MR 2764786, 10.1007/s11425-010-4148-7 Reference: [6] Rao, H., Ruan, H.-J., Wang, Y.: Lipschitz equivalence of Cantor sets and algebraic properties of contraction ratios.Trans. Am. Math. Soc. 364 (2012), 1109-1126. Zbl 1244.28015, MR 2869169, 10.1090/S0002-9947-2011-05327-4 Reference: [7] Wang, B.-W., Wu, J.: Hausdorff dimension of certain sets arising in continued fraction expansions.Adv. Math. 218 (2008), 1319-1339. Zbl 1233.11084, MR 2419924, 10.1016/j.aim.2008.03.006 Reference: [8] Wang, X. H., Wen, S. Y.: Doubling measures on Cantor sets and their extensions.Acta Math. Hung. 134 (2012), 431-438. Zbl 1265.28001, MR 2886217, 10.1007/s10474-011-0186-z Reference: [9] Wu, J.: On the sum of degrees of digits occurring in continued fraction expansions of Laurent series.Math. Proc. Camb. Philos. Soc. 138 (2005), 9-20. Zbl 1062.11054, MR 2127223, 10.1017/S0305004104008163 Reference: [10] Wu, M.: The singularity spectrum {$f(\alpha)$} of some Moran fractals.Monatsh. Math. 144 (2005), 141-155. Zbl 1061.28005, MR 2123961, 10.1007/s00605-004-0254-3 .

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