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

Das, Tushar; Fishman, Lior; Simmons, David; Urbański, Mariusz

doi:10.1007/s00029-017-0324-8

Cited by 15 publications

(22 citation statements)

References 23 publications

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“…These facts are proven in [, Theorem 1.9; , Theorem 2.3], respectively. Recently, the concept of friendly measures has been generalised even further to the notion of quasi‐decaying measures, see .…”

Section: The General Setup and Main Problemsmentioning

confidence: 99%

Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points

Beresnevich

Ghosh

Simmons

et al. 2018

J. London Math. Soc.

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The overall aim of this note is to initiate a "manifold" theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural "manifold" strengthening of Sullivan's logarithmic law for geodesics. The general setup and main problemsThe classical results of Diophantine approximation, in particular those from the onedimensional theory, have natural counterparts and extensions in the hyperbolic space setting. In this setting, instead of approximating real numbers by rationals, one approximates the limit points of a fixed Kleinian group G by points in the orbit (under the group) of a distinguished limit point y. Beardon & Maskit [4] have shown that the geometry of the group is reflected in the approximation properties of points in the limit set.Unless stated otherwise, in what follows G denotes a nonelementary, geometrically finite Kleinian group acting on the unit ball model (B d+1 , ρ) of (d + 1)-dimensional hyperbolic space with metric ρ derived from the differential dρ = 2|dx|/(1−|x| 2 ). Thus, G is a discrete subgroup of Möb(B d+1 ), the group of orientation-preserving Möbius *

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Section: The General Setup and Main Problemsmentioning

confidence: 99%

Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points

Beresnevich

Ghosh

Simmons

et al. 2018

J. London Math. Soc.

Self Cite

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“…The natural candidate is what we call the Schubert closure of M, which is the intersection of the pencils containing M. It is an algebraic variety, which is in general bigger than the Zariski closure of M. In the classical setting of submanifolds of M m,n (R) this is the same space as the space H(M) considered by Beresnevich, Kleinbock and Margulis in [BKM15, 7.2]. The Plücker closure considered here and in [DFSU15] is in general smaller than the Schubert closure.…”

Section: Introductionmentioning

confidence: 98%

“…Theorem 1.4 has been established independently in the recent work of Das, Fishman, Simmons and Urbanski, see [DFSU15,Theorem 1.9]. In this work the authors introduce a very general class of measures, which is invariant under measure automorphisms and encompasses many fractal measures of dynamical origin, for which Theorem 1.4 is shown to hold, see the discussion after [DFSU15, Definition 1.2].…”

Section: Introductionmentioning

confidence: 99%

Diophantine approximation on matrices and Lie groups

et al. 2018

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Abstract. We study the general problem of extremality for metric diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over Q, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest. In particular we prove that the diophantine exponent of rational nilpotent Lie groups exists and is a rational number, which we determine explicitly in terms of representation theoretic data.

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“…For more than a decade now, as part of the burgeoning study of Diophantine properties of fractal sets and measures [5,9,10,13,19], there has been a growing interest in computing the Hausdorff dimension of the intersection of BA d with various fractal sets. Since BA d has full dimension, one expects its intersection with any fractal set J ⊆ R d to have the same dimension as J , and this can be proven for certain broad classes of fractal sets J , see e.g.…”

Section: T Das Et Almentioning

confidence: 99%

“…Now let τ ∈ E * = ∞ n=0 E n be a word of fixed length (independent of N ) such that 5) and consider the diagonal IFS = N = (ψ ω ) ω∈T 0 , where for each ω ∈ T 0 we write…”

Section: Proof Of the Main Theoremsmentioning

confidence: 99%

Badly approximable points on self-affine sponges and the lower Assouad dimension

Das¹,

Fishman²,

Simmons³

et al. 2017

Ergod. Th. Dynam. Sys.

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Abstract. We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford-McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.Fix d ∈ N. Dirichlet's theorem in Diophantine approximation states that, for all x ∈ R d , there exist infinitely many rational points p/q ∈ Q d such thatA point x ∈ R d is said to be badly approximable if this inequality cannot be improved by more than a constant, i.e. if there exists a constant c > 0 such that, for any rational

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

Cited by 15 publications

References 23 publications

Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points

Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points

Diophantine approximation on matrices and Lie groups

Badly approximable points on self-affine sponges and the lower Assouad dimension

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