Diophantine approximation on matrices and Lie groups

Aka, Menny; Breuillard, Emmanuel; Rosenzweig, Lior; Saxcé, Nicolas de

doi:10.1007/s00039-018-0436-0

Cited by 14 publications

(32 citation statements)

References 39 publications

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“…See [1,2] for a recent discussion of Diophantine properties in groups and related problems. There, the definition is more general, replacing the separation in (1.1) with a negative power of the cardinality of the n-ball in the word metric in the group generated by A.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…There, the definition is more general, replacing the separation in (1.1) with a negative power of the cardinality of the n-ball in the word metric in the group generated by A. In [2] a semi-simple Lie group G is called Diophantine, if almost every k elements of G, chosen independently at random according to the Haar measure, together with their inverses, form a Diophantine set in G. Gamburd et al [15] conjectured that SU 2 (R) is Diophantine. More generally, it is conjectured that semi-simple Lie groups are Diophantine.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

Solomyak

Takahashi

2020

International Mathematics Research Notices

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We prove that almost every finite collection of matrices in GL d (R) and SL d (R) with positive entries is Diophantine. Next we restrict ourselves to the case d = 2. A finite set of SL 2 (R) matrices induces a (generalized) iterated function system on the projective line RP 1 . Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

Solomyak

Takahashi

2020

International Mathematics Research Notices

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“…2 When the open set U is not explicitly mentioned, we assume that it is all of R d ; otherwise we say that μ is absolutely decaying, friendly, etc. "relative to U ".…”

Section: Four Conditions Which Imply Strong Extremalitymentioning

confidence: 99%

“…But P is asymptotic to the maximum of the coefficients of P, which implies that P (2) × P . On the other hand, R (2) × f by (3.9), so overall we have f (2) × f . Applying the mean value inequality yields (3.8).…”

Section: )mentioning

confidence: 99%

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures

Das

Fishman

Simmons

et al. 2017

Sel. Math. New Ser.

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We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis ('98) resolving Sprindžuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss ('04), hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, of the Patterson-Sullivan measures of all nonplanar geometrically finite groups, and of the Gibbs measures (including conformal measures) of infinite iterated function systems. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW's sufficient conditions for extremality. In Part I, we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, thus proving a conjecture of KLW. We also prove the "inherited exponent of irrationality" version of this theorem, describing the relationship between the Diophantine properties of certain subspaces of the space of matrices and measures supported on these subspaces. In subsequent papers, we exhibit numerous examples of quasi-decaying measures, in support of the thesis that "almost any measure from dynamics and/or fractal geometry is quasi-decaying". We also discuss examples of non-extremal measures coming from dynamics, illustrating where the theory must halt.

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“…Their work was later extended from R n to the space M m×n (R) of m×n real matrices (see e.g. [KMW10][BKM15] [ABRdS18]).…”

Section: Introductionmentioning

confidence: 99%

Equidistribution of expanding translates of curves and Diophantine approximation on matrices

Yang

2020

Invent. math.

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We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space G/Γ of a semisimple algebraic group G. We define two families of algebraic subvarieties of the associated partial flag variety G/P , which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of m × n real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved.The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.

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Diophantine approximation on matrices and Lie groups

Cited by 14 publications

References 39 publications

Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures

Equidistribution of expanding translates of curves and Diophantine approximation on matrices

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