On Nikodym-type sets in high dimensions
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.
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.
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.
Antti Käenmäki, Markku Vilppolainen (2008)
Fundamenta Mathematicae
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It is well known that the open set condition and the positivity of the t-dimensional Hausdorff measure are equivalent on self-similar sets, where t is the zero of the topological pressure. We prove an analogous result for a class of Moran constructions and we study different kinds of Moran constructions in this respect.
Veerman, J.J.P., Stošić, B.D. (2000)
Experimental Mathematics
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Michał Rams (2006)
Bulletin of the Polish Academy of Sciences. Mathematics
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We estimate from above and below the Hausdorff dimension of SRB measure for contracting-on-average baker maps.
Lu-ming Shen (2010)
Acta Arithmetica
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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).
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...
Jaroslav Hančl, Radhakrishnan Nair, Lukáš Novotný, Jan Šustek (2012)
Acta Arithmetica
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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.
Yan-Yan Liu, Jun Wu (2001)
Acta Arithmetica
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Roy O. Davies (1979)
Colloquium Mathematicae
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Giuseppe Devillanova, Sergio Solimini (2007)
Rendiconti del Seminario Matematico della Università di Padova
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F. Przytycki, M. Urbański (1989)
Studia Mathematica
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Igudesman, K. (2003)
Lobachevskii Journal of Mathematics
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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.
Huw Jones (2001)
Acta Arithmetica
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L. B. Jonker, J. J. P. Veerman (2002)
Fundamenta Mathematicae
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The basic question of this paper is: If you consider two iterated function systems close to each other in an appropriate topology, are the dimensions of their respective invariant sets close to each other? It is well known that the Hausdorff dimension (and Lebesgue measure) of the invariant set does not depend continuously on the iterated function system. Our main result is that (with a restriction on the "non-conformality" of the transformations) the Hausdorff dimension is a lower semicontinuous...