Ultrametric logarithm laws, II

Athreya, Jayadev S.; Ghosh, Anish; Prasad, Amritanshu

doi:10.1007/s00605-012-0376-y

Cited by 26 publications

(41 citation statements)

References 44 publications

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“…It depends on constructing Ahlfors regular subsets of the uniformly radial limit set with dimension arbitrarily close to the Poincaré exponent δ (Theorem 5.9); in this way, we also generalize the seminal theorem of C. J. Bishop and P. W. Jones [10] which states that the radial and uniformly radial limit sets Λ r (G) and Λ ur (G) each have dimension δ. Theorem 5.9 will appear in a stronger form in [26,Theorem 1.2.1], whose authors include the second-and third-named authors of this paper. 2 Finally, as an application of our generalization of Khinchin's theorem we show that for any group acting on a proper geodesic hyperbolic metric space, the uniformly radial limit set has zero Patterson-Sullivan 1 Although they do not fall into our framework, let us mention in passing Hersonsky and Paulin's paper [45], which estimates the dimension of the set of geodesic rays in a pinched Hadamard manifold X which return exponentially close to a given point of X infinitely often, and the two papers [4,53], which generalize Khinchin's theorem in a somewhat different direction than us. 2 Let us make it clear from the start that although we cite the preprint [26] frequently for additional background, the results which we quote from [26] are not used in any of our proofs.…”

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confidence: 94%

Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces

2018

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In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic '76 paper to more recent results of Hersonsky and Paulin ('02, '04, '07). Concrete examples of situations we consider which have not been considered before include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of any element of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones ('97) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson-Sullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.

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confidence: 94%

Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces

2018

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“…V. Sprindžuk established the positive characteristic analogue of Mahler's conjecture in [49]. We refer the reader to [15,36] for general surveys and to [3,19,21,23,25,34,35,37] for some of the recent developments. Sprindžuk's conjecture over a local field of positive characteristic was settled by Anish Ghosh in [22].…”

Section: Introductionmentioning

confidence: 99%

Dirichlet improvability for $S$-numbers

Das¹,

Ganguly²

2022

Preprint

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We study the problem of improving Dirichlet's theorem of metric Diophantine approximation in the S-adic setting. Our approach is based on translation of the problem related to Dirichlet improvability into a dynamical one, and the main technique of our proof is the S-adic version of quantitative nondivergence estimate due to D. Y. Kleinbock and G. Tomanov. The main result of this paper can be regarded as the number field version of earlier works of D. Y. Kleinbock and B. Weiss, and of the second named author and Anish Ghosh.

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“…V. Sprindžuk proved the analogue of Mahler's conjectures and some transference principles for function fields (see [24]). The reader is being referred to [5,21] for general surveys and also to [2,20,14,13,7] (for a necessarily incomplete set of references) to see the state of the art of some of the recent developments.…”

Section: Introductionmentioning

confidence: 99%