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

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

doi:10.1017/etds.2017.42

Cited by 11 publications

(13 citation statements)

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“…Here there is a well-developed connection between estimating the Hausdorff dimension of sets of badly approximable numbers and the lower dimension, see work of Das, Fishman, Simmons and Urbański [45]. Finally, in Section 14.3 we discuss an elegant use of the Assouad dimension in problems of definability of the integers due to Hieronymi and Miller [131].…”

Section: Almost Bi-lipschitz Embeddingsmentioning

confidence: 99%

“…The setting where F is a manifold is motivated by work of Davenport [50] 2 and has developed into a broad research area, see [27] for some background. The second context, where F is a fractal set, has gained a lot of attention in the literature, see [45,46,171]. It is in this second context that a surprising connection to the lower dimension has emerged.…”

Section: Diophantine Approximationmentioning

confidence: 99%

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Assouad Dimension and Fractal Geometry

Fraser

2020

111

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Dimension theory of orthogonal projections vi Contents 10.2.2 The Assouad dimension of orthogonal projections 10.2.3 An application to the box dimensions of projections 10.2.4 The lower dimension and projections 10.3 Slices and intersections 11 Two famous problems in geometric measure theory 11.1 Distance sets 11.2 Kakeya sets 12 Conformal dimension 12.1 Lowering the Assouad dimension by quasi-symmetry PART THREE APPLICATIONS 13 Applications in embedding theory 13.1 Assouad's embedding theorem 13.1.1 Doubling and uniformly perfect metric spaces 13.1.2 Assouad's embedding theorem 13.2 The spiral winding problem 13.3 Almost bi-Lipschitz embeddings 14 Applications in number theory 14.1 Arithmetic progressions 14.1.1 Discrete Kakeya sets 14.2 Diophantine approximation 14.3 Definability of the integers 15 Applications in probability theory 15.1 Uniform dimension results 15.2 Dimensions of random graphs 16 Applications in functional analysis 16.1 Hardy inequalities 16.2 L p → L q bounds for spherical maximal operators 16.3 Connection with L p -norms 17 Future directions 17.1 Finite stability of modified lower dimension 17.2 Dimensions of measures 17.3 Weak tangents 17.4 Further questions of measurability 17.5 IFS attractors Contents vii 17.6 Random sets 17.7 General behaviour of the Assouad spectrum 17.8 Projections 17.9 Distance sets 17.10 The Hölder mapping problem and dimension 17.11 Dimensions of graphs References List of notation Index PART ONE THEORY 1 Fractal geometry and dimension theoryIn this introductory chapter we briefly discuss the history and development of fractal geometry and dimension theory. We introduce and motivate some important concepts such as Hausdorff and box dimension. As part of this discussion we encounter covers and packings, which are central notions in dimension theory, and introduce the dimension theory of measures. The emergence of fractal geometryA fractal can be described as an object which exhibits interesting features on a large range of scales, see Figure 1.1. In pure mathematics, the Sierpiński triangle, the middle third Cantor set, the boundary of the Mandelbrot set, and the von Koch snowflake are archetypal examples and, in 'real life', examples include the surface of a lung, the horizon of a forest, and the distribution of stars in the galaxy. The fractal story began in the nineteenth century with the appearance of a multitude of strange examples exhibiting what we now understand as fractal phenomena. These included Weierstrass' example of a continuous nowhere differentiable function, Cantor's construction of an uncountable set with zero length, and Brown's observations on the path taken by a piece of pollen suspended in water (Brownian motion). During the first half of the twentieth century the mathematical foundations for fractal geometry were laid down by, for example, Besicovitch, Bouligand, Hausdorff, Julia, Marstrand and Sierpiński, and the theory was unified and popularised by the extensive writings of Mandelbrot in the 1970s, for example [197]. It was Mandelbrot who co...

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Section: Almost Bi-lipschitz Embeddingsmentioning

confidence: 99%

Section: Diophantine Approximationmentioning

confidence: 99%

Assouad Dimension and Fractal Geometry

Fraser

2020

111

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“…A stronger condition than the OSC is the so called strong OSC which assumes the existence of an OSC set U such that U ∩ K = ∅ . It was shown by Schief [22] that for similarity 3 IFSs the OSC and the strong OSC are equivalent.…”

Section: Iterated Function Systemsmentioning

confidence: 99%

“…In [11], Furstenberg considered the process of repeated magnification of a set as a dynamical system called a CP-process, and used methods from ergodic theory to get results regarding the Hausdorff dimension of the image of the set under linear transformations, whenever the collection of microsets satisfies certain conditions (see section 5 for more details). As another example, certain properties of the collection of microsets of a given compact set provide interesting information regarding the intersection of the set with winning sets for a variation of Schmidt's game called the hyperplane absolute game ( [2], [3]).…”

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confidence: 99%

On microsets, Assouad dimension and lower dimension of random fractals, and Furstenberg's homogeneity

Dayan¹

2022

Preprint

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We study the collection of microsets of randomly constructed fractals, which in this paper, are referred to as Galton-Watson fractals. This is a model that generalizes Mandelbrot percolation, where Galton-Watson trees (whose offspring distribution is not necessarily binomial) are projected to R d by a coding map which arises from an iterated function system (IFS) of similarity maps. We show that for such a random fractal E, whenever the underlying IFS satisfies the open set condition, almost surely the Assouad dimension of E is the maximal Hausdorff dimension of a set in supp (E), the lower dimension is the smallest Hausdorff dimension of a set in supp (E), and every value in between is the Hausdorff dimension of some microset of E.In order to obtain the above, we first analyze the relation between the collection of microsets of a (deterministic) set, and certain limits of subtrees of an appropriate coding tree for that set.The results of this analysis are also applied, with the required adjustments, to gain some insights on Furstenberg's homogeneity property. Corresponding to a discussion in [15], we define a weaker property than homogeneity and show that for self-homothetic sets in R whose Hausdorff dimension is smaller than 1, it is equivalent to the weak separation condition.

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“…The above lemma was originally proved by Fishman [, Theorem 3.1] but shorter proofs have appeared in the literature since then, see, for example, [, Proposition 2.5; , Lemma 5.8]. The difference between these proofs and the one appearing below is that the one below emphasises the connection with the ‘hands‐on’ technique for producing Cantor sets with a certain property.…”

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

Cited by 11 publications

References 23 publications

Assouad Dimension and Fractal Geometry

Assouad Dimension and Fractal Geometry

On microsets, Assouad dimension and lower dimension of random fractals, and Furstenberg's homogeneity

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

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