Hausdorff dimension of the set of singular pairs

Cheung, Yitwah

doi:10.4007/annals.2011.173.1.4

Cited by 62 publications

(99 citation statements)

References 28 publications

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“…In a remarkable article, the Hausdorff dimension of Sing(2) was determined by Cheung [Che11]. This result was extended by Cheung and Chevallier in [CC16] to higher dimensions.…”

Section: Introductionmentioning

confidence: 91%

Singular Vectors on Fractals and Projections of Self-similar Measures

Khalil

2020

Geom. Funct. Anal.

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Singular vectors are those for which the quality of rational approximations provided by Dirichlet's Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular vectors lying on self-similar fractals in R d satisfying the open set condition. The bound is in terms of quantities which are closely tied to Frostman exponents of projections of the Hausdorff measure supported on the fractal. Our bound is optimal in the sense that it agrees with the exact dimension of singular vectors obtained by Cheung and Chevallier when the fractal is trivial (i.e. has non-empty interior). As a corollary, we show that if the fractal is the product of 2 copies of Cantor's middle thirds set or the attractor of a planar homogeneous irrational IFS, then the upper bound is 2/3 the dimension of the fractal. This addresses the upper bound part of a question raised by Bugeaud, Cheung and Chevallier. We apply our method in the setting of translation flows on flat surfaces to show that the dimension of non-uniquely ergodic directions belonging to a fractal is at most 1/2 the dimension of the fractal.

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“…In a remarkable article, the Hausdorff dimension of Sing(2) was determined by Cheung [Che11]. This result was extended by Cheung and Chevallier in [CC16] to higher dimensions.…”

Section: Introductionmentioning

confidence: 91%

Singular Vectors on Fractals and Projections of Self-similar Measures

Khalil

2020

Geom. Funct. Anal.

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“…When the quadric X has Q-rank at least 2, it is natural to expect that there exist some nontrivial singular points. It might then be interesting to compute the Hausdorff dimension of the set of singular points on X, similarly to what has been done in [7,8] for Diophantine approximation in the Euclidean space.…”

Section: Further Directions and Open Problemsmentioning

confidence: 99%

Rational approximation on quadrics: A simplex lemma and its consequences

Kleinbock

Saxcé

2019

Enseign. Math.

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“…Davenport and Schmidt in the 1970s showed that

Sing (d)

is a set of

d

‐dimensional Lebesgue measure zero. Recently, Cheung and Chevallier , building on the spectacular

d = 2

work of Cheung , have shown that

Sing (d)

has Hausdorff dimension

d^{2} / false(d + 1 false)

.…”

Section: The General Setup and Main Problemsmentioning

confidence: 99%

“…It is not difficult to see that the set Sing(d) of singular vectors contains every rational hyperplane in R d and thus its Hausdorff dimension is between d − 1 and d. In the case d = 1, a nifty argument (which we will utilise) due to Khintchine [24] shows that a real number is singular if and only if it is rational; that is, Sing(1) = Q. Davenport and Schmidt [20] in the 1970s showed that Sing(d) is a set of d-dimensional Lebesgue measure zero. Recently, Cheung and Chevallier [16], building on the spectacular d = 2 work of Cheung [15], have shown that Sing(d) has Hausdorff dimensiond 2 /(d + 1). † The stabiliser Gp = {g ∈ G : g(p) = p} of a parabolic fixed point p is an infinite group that contains a free abelian subgroup of finite index.…”

Section: A Dirichlet-type Theorem and Singular Subsets Of λmentioning

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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Hausdorff dimension of the set of singular pairs

Cited by 62 publications

References 28 publications

Singular Vectors on Fractals and Projections of Self-similar Measures

Singular Vectors on Fractals and Projections of Self-similar Measures

Rational approximation on quadrics: A simplex lemma and its consequences

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

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