Ergodic Theory on Homogeneous Spaces and Metric Number Theory

Kleinbock, Dmitry

doi:10.1007/978-0-387-30440-3_180

Cited by 4 publications

(5 citation statements)

References 66 publications

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“…[5], [20], [6], [10] [14], [22], [17]) that in many kind of (more or less) hyperbolic or "fast" mixing systems, various sequences of geometrically nice sets have the BC property. The kind of sets which are interesting to be considered in this kind of problems are usually decreasing sequences of balls with the same center (these are also called shrinking targets, this approach has relations with the theory of approximation speed, see [8], [19]) or cylinders.…”

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confidence: 99%

The dynamical Borel-Cantelli lemma and the waiting time problems

Galatolo

Kim

2007

Indagationes Mathematicae

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We investigate the connection between the dynamical Borel-Cantelli and waiting time results. We prove that if a system has the dynamical Borel-Cantelli property, then the time needed to enter for the first time in a sequence of small balls scales as the inverse of the measure of the balls. Conversely if we know the waiting time behavior of a system we can prove that certain sequences of decreasing balls satisfies the Borel-Cantelli property. This allows to obtain Borel-Cantelli like results in systems like axiom A and generic interval exchanges.

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confidence: 99%

The dynamical Borel-Cantelli lemma and the waiting time problems

Galatolo

Kim

2007

Indagationes Mathematicae

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“…Note that for a fixed H the number of different pairs (a 1 , a 2 ) is no greater than 4H. By (22) there are O(H) possibilities for a 0 if (a 1 , a 2 ) are fixed. Therefore an appropriate s-volume sum for C n will be…”

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confidence: 98%

“…Diophantine approximation on manifolds has been an extremely active research area over the past 10 years or so. Rather than describe the activity in detail, we refer the reader to research articles [3,4,8,10,15,23,29] and the surveys [6,22,25]. Nevertheless, it is worth singling out the pioneering work of Kleinbock and Margulis [23] in which the fundamental Baker-Sprindzuk conjecture is established.…”

Section: Introductionmentioning

confidence: 99%

Inhomogeneous Diophantine approximation on curves and Hausdorff dimension

Badziahin

2010

Advances in Mathematics

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The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in R n akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the fundamental homogeneous theorems of R.C. Baker (1978), Dodson, Dickinson (2000) and Beresnevich, Bernik, Kleinbock, Margulis (2002). In the case of planar curves, the complete Hausdorff dimension theory is developed.

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“…For several decades, tools from dynamical systems, and in particular ergodic theory, have been used to derive arithmetic and number theoretic, in particular Diophantine approximation results, see for instance the works of Furstenberg, Margulis, Sullivan, Dani, Kleinbock, Clozel, Oh, Ullmo, Lindenstrauss, Einsiedler, Michel, Venkatesh, Marklof, Green-Tao, Elkies-McMullen, Ratner, Mozes, Shah, Gorodnik, Ghosh, Weiss, Hersonsky-Paulin, Parkkonen-Paulin and many others, and the references [Kle2,Lin,Kle1,AMM,Ath,GorN,EiW,PaP5].…”

Section: Introductionmentioning

confidence: 99%

A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature

Parkkonen

Paulin

2017

Lecture Notes in Mathematics

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This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by quadratic irrational ones, and we discuss various results on the equidistribution in R, C and in the Heisenberg groups of arithmetically defined points. We explain how these results are consequences of equidistribution and counting properties of common perpendiculars between locally convex subsets in negatively curved orbifolds, proven using dynamical and ergodic properties of their geodesic flows. This exposition is based on lectures at the conference "Chaire Jean Morlet: Géométrie et systèmes dynamiques", at the CIRM, Luminy, 2014. We thank B. Hasselblatt for his strong encouragements to write this survey.

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Ergodic Theory on Homogeneous Spaces and Metric Number Theory

Cited by 4 publications

References 66 publications

The dynamical Borel-Cantelli lemma and the waiting time problems

The dynamical Borel-Cantelli lemma and the waiting time problems

Inhomogeneous Diophantine approximation on curves and Hausdorff dimension

A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature

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