On the dimensions of certain incommensurably constructed sets.
Veerman, J.J.P., Stošić, B.D. (2000)
Experimental Mathematics
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Veerman, J.J.P., Stošić, B.D. (2000)
Experimental Mathematics
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W. Kulpa (1972)
Colloquium Mathematicae
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Themis Mitsis (2004)
Studia Mathematica
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We prove that the complement of a higher-dimensional Nikodym set must have full Hausdorff dimension.
R. Duda (1979)
Colloquium Mathematicae
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James R. Lee, Manor Mendel, Mohammad Moharrami (2012)
Fundamenta Mathematicae
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For every ε > 0, any subset of ℝⁿ with Hausdorff dimension larger than (1-ε)n must have ultrametric distortion larger than 1/(4ε).
R. Đorđević (1989)
Matematički Vesnik
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Balázs Bárány (2009)
Fundamenta Mathematicae
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We investigate the properties of the Hausdorff dimension of the attractor of the iterated function system (IFS) {γx,λx,λx+1}. Since two maps have the same fixed point, there are very complicated overlaps, and it is not possible to directly apply known techniques. We give a formula for the Hausdorff dimension of the attractor for Lebesgue almost all parameters (γ,λ), γ < λ. This result only holds for almost all parameters: we find a dense set of parameters (γ,λ) for which the Hausdorff...
Lu-ming Shen (2010)
Acta Arithmetica
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Simon Baker, Karma Dajani, Kan Jiang (2015)
Fundamenta Mathematicae
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Let K ⊆ ℝ be the unique attractor of an iterated function system. We consider the case where K is an interval and study those elements of K with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams (1988). The theory of Mauldin and Williams then provides...
Józef Myjak, Ryszard Rudnicki (2007)
Bollettino dell'Unione Matematica Italiana
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A relationship between the information dimension and the average dimension of a measure is given. Properties of the average dimension are studied.
L. Polkowski (1985)
Colloquium Mathematicae
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Igudesman, K. (2003)
Lobachevskii Journal of Mathematics
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Jaroslav Hančl, Radhakrishnan Nair, Lukáš Novotný, Jan Šustek (2012)
Acta Arithmetica
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Myjak, Józef (2005)
Abstract and Applied Analysis
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Michael G. Charalambous (2006)
Fundamenta Mathematicae
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We construct in ZFC a cosmic space that, despite being the union of countably many metrizable subspaces, has covering dimension equal to 1 and inductive dimensions equal to 2.
Juan B. Sancho de Salas, M.ª Teresa Sancho de Salas (1988)
Extracta Mathematicae
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C. Das, G. Patel (1988)
Matematički Vesnik
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Quansheng Liu (1993)
Publications mathématiques et informatique de Rennes
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T. W. Körner (2008)
Studia Mathematica
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There is no non-trivial constraint on the Hausdorff dimension of sums of a set with itself.
Cutler, C.D. (1988)
International Journal of Mathematics and Mathematical Sciences
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Guifeng Huang, Lidong Wang (2014)
Annales Polonici Mathematici
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A solution of the Feigenbaum functional equation is called a Feigenbaum map. We investigate the likely limit set (i.e. the maximal attractor in the sense of Milnor) of a non-unimodal Feigenbaum map, prove that it is a minimal set that attracts almost all points, and then estimate its Hausdorff dimension. Finally, for every s ∈ (0,1), we construct a non-unimodal Feigenbaum map with a likely limit set whose Hausdorff dimension is s.
Ondřej Zindulka (2012)
Fundamenta Mathematicae
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We prove that each analytic set in ℝⁿ contains a universally null set of the same Hausdorff dimension and that each metric space contains a universally null set of Hausdorff dimension no less than the topological dimension of the space. Similar results also hold for universally meager sets. An essential part of the construction involves an analysis of Lipschitz-like mappings of separable metric spaces onto Cantor cubes and self-similar sets.