Generalised Cantor sets and the dimension of products

Olson, Eric J.; Robinson, James C.; Sharples, Nicholas

doi:10.1017/s0305004115000584

Cited by 15 publications

(20 citation statements)

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“…We begin, in Section 2, by introducing our terminology and notation. There we also derive formulas for the (Lower) Assouad dimensions of the associated sets C a , generalizing the formulas found in [15] and [20] for the special case of central Cantor sets. These formulas will be very useful for the proofs given later in the paper.…”

Section: Introductionmentioning

confidence: 76%

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Assouad dimensions of complementary sets

García¹,

Hare²,

Mendivil³

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Given a non-negative, decreasing sequence a with sum 1, we consider all the closed subsets of [0, 1] such that the lengths of their complementary open intervals are given by the terms of a, the so-called complementary sets. In this paper we determine the almost sure value of the Φ-dimensions of these sets given a natural model of randomness. The Φ-dimensions are intermediate Assouad-like dimensions which include the Assouad and quasi-Assouad dimensions as special cases. The answers depend on the size of Φ, with one size behaving like the Assouad dimension and the other, like the quasi-Assouad dimension.

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Section: Introductionmentioning

confidence: 76%

“…Similar arguments show that the same formula holds if C{r j } is a central Cantor set. Previously, the Assouad dimension formula ( 6) was obtained for central Cantor sets, using other methods, by Olson et al in [20]; see also Li et al [15].…”

Section: 2mentioning

confidence: 99%

Assouad dimensions of complementary sets

García¹,

Hare²,

Mendivil³

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…The fact that the Assouad dimension is subadditive on products, that is, dim [192,Theorem A.5], see also [240,Lemma 9] and [88,Proposition 2.1]. Sharpness of these product formulae was considered in [224,223].…”

Section: Productsmentioning

confidence: 99%

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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“…Assouad's original motivation was to study embedding problems, a subject where the Assouad dimension is still playing a fundamental rôle, see [Ol, OR, R]. The concept has also found a home in other areas of mathematics, including the theory of quasi-conformal mappings [H, L, MT], and more recently it is gaining substantial attention in the literature on fractal geometry [K,M,O,Fr3,LLMX,FHOR,ORS]. It is also worth noting that, due to its intimate relationship with tangents, it has always been present, although behind the scenes, in the pioneering work of Furstenberg on micro-sets and the related ergodic theory which goes back to the 1960s, see [Fu].…”

Section: The Assouad Dimensionmentioning

confidence: 99%

The Assouad dimension of randomly generated fractals

Fraser¹,

Miao²,

Troscheit³

2016

Ergod. Th. Dynam. Sys.

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We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a common phenomenon in all of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.Mathematics Subject Classification 2010: primary: 28A80, 60J80; secondary: 37C45, 54E52, 82B43.

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Generalised Cantor sets and the dimension of products

Cited by 15 publications

References 15 publications

Assouad dimensions of complementary sets

Assouad dimensions of complementary sets

Assouad Dimension and Fractal Geometry

The Assouad dimension of randomly generated fractals

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