On fractal measures and diophantine approximation

By Dmitry Kleinbock, Elon Lindenstrauss, Barak Weiss. 2005

178 citations recorded by the source corpus.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/784ae30a2f5a61bf88ceb4b4823d637b3a6a26c6

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In Condensed Matter Physics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

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.”
An extension of quantitative nondivergence and applications to Diophantine exponents (2005, methodology): “Therefore in all the problems related to the extremality of manifolds/measures, it makes no dierence whether to interpret points of Rn as row vectors (linear forms) or as column vectors (the latter set-up was employed in [ KLW , KW1{2] and is usually referred to as simultaneous approximation, hence ”
Rational points near manifolds and metric Diophantine approximation (2009, background): “The work of Kleinbock and Margulis has also dealt with the far more delicate multiplicative case known as the Baker-Sprindžuk conjecture and led to a surge of activity that led to establishing the extremality of various classes of manifolds and sets; see, for example, [38], [39], [40], [42].”
Badly approximable vectors on fractals (2005, methodology): “Assuming that f’ig is irreducible, it was proved in [ KLW , x8] that is absolutely decaying,”
Badly approximable points on manifolds (2013, background): “Beyond nondegenerate manifolds, it would be interesting to obtain generalisations of Theorems 1 and 2 for friendly measures as defined in [32] as well as for affine subspaces of R and their submanifolds—see [31] for a related context.”
Badly approximable systems of affine forms, fractals, and Schmidt games (2009, background): “(See, for example, [11], [8], and [15] for additional details on these fractals.”
Schmidt’s game on fractals (2009, methodology): “Following [ KLW ](Theorem 2.3, Lemmas 8.2 and 8.3), there exist C and a such that µ is absolutely (C,a)-decaying (see Remark 2) on any ball of radius r = 1 centered in supp(µ).”
DIRICHLET'S THEOREM ON DIOPHANTINE APPROXIMATION AND HOMOGENEOUS FLOWS (2006, methodology): “We need to introduce some more notation in order to state a theorem from [14].”