The Experts below are selected from a list of 20805 Experts worldwide ranked by ideXlab platform
Yitwah Cheung - One of the best experts on this subject based on the ideXlab platform.
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Hausdorff Dimension of singular vectors
Duke Mathematical Journal, 2016Co-Authors: Yitwah Cheung, Nicolas ChevallierAbstract:We prove that the set of singular vectors in R, d ≥ 2, has Hausdorff Dimension d d+1 and that the Hausdorff Dimension of the set of e-Dirichlet improvable vectors in R d is roughly d 2 d+1 plus a power of e between d 2 and d. As a corollary, the set of divergent trajectories of the flow by diag(e, . . . , e, e−dt) acting on SLd+1(R)/ SLd+1(Z) has Hausdorff coDimension d d+1 . These results extend the work of the first author in [6].
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Hausdorff Dimension of the set of singular pairs
Annals of Mathematics, 2011Co-Authors: Yitwah CheungAbstract:In this paper we show that the Hausdorff Dimension of the set of singular pairs is 4 3 . We also show that the action of diag(e t , e t , e ―2t ) on SL 3 R/SL 3 Z admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates, answering a question of A. N. Starkov. As a by-product of the analysis, we obtain a higher-Dimensional generalization of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the technique used to compute Hausdorff Dimension, we reprove a result of I. J. Good asserting that the Hausdorff Dimension of the set of real numbers with divergent partial quotients is 1 2 .
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Hausdorff Dimension of the set of singular pairs
arXiv: Dynamical Systems, 2007Co-Authors: Yitwah CheungAbstract:In this paper we show that the Hausdorff Dimension of the set of singular pairs is 4/3. We also show that the action of diag(e^t,e^t,e^{-2t}) on SL(3,R)/SL(3,Z) admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates, answering a question of A.N. Starkov. As a by-product of the analysis, we obtain a higher Dimensional generalisation of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the techniques used to compute Hausdorff Dimension, we show that the set of real numbers with divergent partial quotients has Hausdorff Dimension 1/2.
Yan-fang Zhang - One of the best experts on this subject based on the ideXlab platform.
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A LOWER BOUND OF TOPOLOGICAL Hausdorff Dimension OF FRACTAL SQUARES
Fractals, 2020Co-Authors: Yan-fang ZhangAbstract:Given an integer [Formula: see text] and a digit set [Formula: see text], there is a self-similar set [Formula: see text] satisfying the set equation [Formula: see text]. This set [Formula: see text] is called a fractal square. By studying the line segments contained in [Formula: see text], we give a lower estimate of the topological Hausdorff Dimension of fractal squares. Moreover, we compute the topological Hausdorff Dimension of fractal squares whose nontrivial connected components are parallel line segments, and introduce the Latin fractal squares to investigate the question when the topological Hausdorff Dimension of a fractal square coincides with its Hausdorff Dimension.
Márton Elekes - One of the best experts on this subject based on the ideXlab platform.
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A new fractal Dimension: The topological Hausdorff Dimension
Advances in Mathematics, 2015Co-Authors: Richárd Balka, Zoltán Buczolich, Márton ElekesAbstract:We introduce a new concept of Dimension for metric spaces, the so-called topological Hausdorff Dimension. It is defined by a very natural combination of the definitions of the topological Dimension and the Hausdorff Dimension. The value of the topological Hausdorff Dimension is always between the topological Dimension and the Hausdorff Dimension, in particular, this new Dimension is a non-trivial lower estimate for the Hausdorff Dimension. We examine the basic properties of this new notion of Dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc. As our first application, we generalize the celebrated result of Chayes, Chayes and Durrett about the phase transition of the connectedness of the limit set of Mandelbrot's fractal percolation process. They proved that certain curves show up in the limit set when passing a critical probability, and we prove that actually ‘thick’ families of curves show up, where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large Hausdorff Dimension. The proof of this is basically a lower estimate of the topological Hausdorff Dimension of the limit set. For the sake of completeness, we also give an upper estimate and conclude that in the non-trivial cases the topological Hausdorff Dimension is almost surely strictly below the Hausdorff Dimension. Finally, as our second application, we show that the topological Hausdorff Dimension is precisely the right notion to describe the Hausdorff Dimension of the level sets of the generic continuous function (in the sense of Baire category) defined on a compact metric space.
Theodore A Slaman - One of the best experts on this subject based on the ideXlab platform.
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irrationality exponent Hausdorff Dimension and effectivization
Monatshefte für Mathematik, 2018Co-Authors: Verónica Becher, Jan Reimann, Theodore A SlamanAbstract:We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff Dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff Dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff Dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff Dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
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irrationality exponent Hausdorff Dimension and effectivization
arXiv: Number Theory, 2016Co-Authors: Verónica Becher, Jan Reimann, Theodore A SlamanAbstract:We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff Dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real less than or equal to 2/a. We show that there is a Cantor-like set with Hausdorff Dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff Dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff Dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
Verónica Becher - One of the best experts on this subject based on the ideXlab platform.
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irrationality exponent Hausdorff Dimension and effectivization
Monatshefte für Mathematik, 2018Co-Authors: Verónica Becher, Jan Reimann, Theodore A SlamanAbstract:We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff Dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff Dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff Dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff Dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
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irrationality exponent Hausdorff Dimension and effectivization
arXiv: Number Theory, 2016Co-Authors: Verónica Becher, Jan Reimann, Theodore A SlamanAbstract:We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff Dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real less than or equal to 2/a. We show that there is a Cantor-like set with Hausdorff Dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff Dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff Dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
