Badly approximable vectors on fractals

By Dmitry Kleinbock, Barak Weiss. 2005

82 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4e455b9305f008ec45666c8376d53af9dbf0008d

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Schmidt’s game, fractals, and orbits of toral endomorphisms (2010, methodology): “The goal of the current work, as well as in several earlier papers [17, 19, 12, 13, 8], is to use measures in order to construct points in their supports with prescribed (dynamical or Diophantine) properties.”
On shrinking targets for Z^m actions on tori (2008, methodology): “Inspired by the works of Kleinbock and Weiss [14] and Kristensen et al [15], we shall deduce Theorem 1 as a simple consequence of a general statement concerning the intersection of BadA with compact subsets of Rn .”
Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces (2013, result): “t to have large Hausdorﬀ dimension without having any closed Ahlfors regular subsets of positive dimension (much less full dimension); in fact it follows from the work of D. Y. Kleinbock and B. Weiss [54] that the set of well approximable numbers forms such a set.9 Remark 1.10. In Theorem 5.9 and in”
On fractal measures and diophantine approximation (2005, background): “In the recent preprint [ KW1 ] the first and third named authors show:”
Badly approximable points on manifolds (2013, background): “It is worth mentioning that the setsBad(r) have been studied at length in all dimensions and for arbitrary collections of weights, see [22,34–36,41].”
The set of badly approximable vectors is strongly C1 incompressible (2011, background): “Many examples of absolutely friendly measures can be found in [17, 18, 32, 29].”
DIRICHLET'S THEOREM ON DIOPHANTINE APPROXIMATION AND HOMOGENEOUS FLOWS (2006, background): “the class of friendly measures is rather large; many examples are described in [KLW, KW1 , Ur1, Ur2, SU].”
Diophantine approximation and Cantor sets (2008, methodology): “Furthermore, metric number theory was used in [7,8,10] to show that K contains badly approximable numbers, that is, numbers ξ for which |ξ − p/q| > c/q2 holds for every rational p/q and some positive constant c.”
On a problem of K. Mahler: Diophantine approximation and Cantor sets (2005, background): “It is well known that Bad ∩ K is non-empty and moreover that dim(Bad ∩ K) = γ—see for example [ 10 ,11].”
Schmidt’s game on fractals (2009, methodology): “Our research follows closely in the footsteps of [KLW], [ KW ] and, consequently, [PV] and [KTV].”
Modified Schmidt games and Diophantine approximation with weights (2008, background): “The following lemma, proved in [11] and generalizing results of C.”
Diophantine approximation and badly approximable sets (2004, background): “tended by Urban´ski [17, 18]. In view of Theorem KLW, the following statement is a simple consequence of Theorem 4. It has also be independently established by Kleinbock & Weiss [9, Theorem 10.3] [10]. Corollary 1 Let {S1,...,Sk} be an irreducible family of contracting self similarity maps of RN”
Assouad Dimension and Fractal Geometry (2020, background): “The second context, where F is a fractal set, has gained a lot of attention in the literature, see [45, 46, 171].”