Limit points badly approximable by horoballs

Mayeda, Dustin; Merrill, Keith

doi:10.1007/s10711-012-9738-9

Cited by 7 publications

(15 citation statements)

References 27 publications

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“…We first require some further notation. For some R and c satisfying (24), assume we are given a class C(n) defined by (25) and a collection of subclasses C(n, m) for 1 m n. We may now introduce our simplification of Theorem 13.…”

Section: Finally Constructmentioning

confidence: 99%

“…Fix B ∈ B(X) and let the parameters R and c satisfy (24). Also, assume that for n ∈ N we have classes C(n) defined by (25), each associated with a collection of subclasses C(n, m) for 1 m n. If for all pairs m, n and for some ǫ > 0 a splitting structure (X, S, U, f ) satisfies q n,m R m(1−ǫ) , then Bad(R, h) is ǫ-Cantor-winning on B with respect to (X, S, U, f ). In particular, we have…”

Section: Finally Constructmentioning

confidence: 99%

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Cantor-winning sets and their applications

Badziahin¹,

Harrap²

2017

Advances in Mathematics

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We introduce and develop a class of Cantor-winning sets that share the same amenable properties as the classical winning sets associated to Schmidt's (α, β)-game: these include maximal Hausdorff dimension, invariance under countable intersections with other Cantor-winning sets and invariance under bi-Lipschitz homeomorphisms. It is then demonstrated that a wide variety of badly approximable sets appearing naturally in the theory of Diophantine approximation fit nicely into our framework. As applications of this phenomenon we answer several previously open questions, including some related to the Mixed Littlewood conjecture and the ×2, ×3 problem.

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Section: Finally Constructmentioning

confidence: 99%

Section: Finally Constructmentioning

confidence: 99%

Cantor-winning sets and their applications

Badziahin¹,

Harrap²

2017

Advances in Mathematics

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“…It turns out to be most convenient to do this by combining a few results which are already known. Namely, a result of Mayeda and Merrill states that the set

{Bad}_{y}

is winning for a different game introduced by McMullen and known as the ‘absolute game’, while the results of Broderick, Fishman, Kleinbock, Reich, and Weiss state that any set winning for the absolute game can be intersected with any sufficiently nice fractal (and in particular any Ahlfors

δ

‐regular fractal) to get a set winning for Schmidt's game (played on that fractal). This immediately implies that

K \cap {boldBad}_{y}

is winning for Schmidt's game (played on

K

) and according to a theorem of Fishman this implies the existence of the family of sets

{F_{s} : s < δ}

described above, and in particular that

dim (K \cap {boldBad}_{y}) ⩾ δ

.…”

Section: Proof Of Theoremmentioning

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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“…11 Remark 1.12. A weaker form of the "moreover" clause was cited in [62] as a private communication from the authors of this paper [62,Lemma 5.2].…”

mentioning

confidence: 99%

“…, ξ ℓ } is a finite set of parabolic points can be deduced from either [35], [81], or [10]. Also note that some further results on winning properties of the set BA ξ can be found in [2], [3], [23], [63], and [62].…”

mentioning

confidence: 99%

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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Limit points badly approximable by horoballs

Cited by 7 publications

References 27 publications

Cantor-winning sets and their applications

Cantor-winning sets and their applications

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

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

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