Properties of quasi-Assouad dimension

García, Ignacio; Hare, Kathryn E.

doi:10.5186/aasfm.2021.4618

Cited by 6 publications

(11 citation statements)

References 21 publications

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“…, The following counter-intuitive result follows from Lemma 3.4.4 and the fact that the Assouad spectrum approaches the quasi-Assouad dimension as θ → 1, see Corollary 3.3.7. We give an alternative direct proof here, following [118]. Proof This result was first proved by García and Hare [118,Proposition 15], and we give their elegant proof, which does not rely on the Assouad spectrum.…”

Section: Relationships Between Dimensionsmentioning

confidence: 94%

“…We give an alternative direct proof here, following [118]. Proof This result was first proved by García and Hare [118,Proposition 15], and we give their elegant proof, which does not rely on the Assouad spectrum. Since dim B F dim qA F always holds, it suffices to prove that if dim B F = 0, then dim qA F = 0.…”

Section: Relationships Between Dimensionsmentioning

confidence: 94%

“…Note that if dim B F = 0, then the Assouad spectrum and quasi-Assouad dimension cannot increase under Lipschitz maps by Corollary 3.4.5, but the Assouad dimension may still increase arbitrarily, see for example [103]. An example showing that the quasi-Assouad dimension can increase under Lipschitz maps can be found in [118,Proposition 12]. Here we show that the Assouad spectrum can also increase and that the upper bound (3.9) is sharp.…”

Section: Further Basic Propertiesmentioning

confidence: 99%

“…With these examples in mind, it might seem hopeless to connect weak tangents to the Assouad spectrum and quasi-Assouad dimension. However, García and Hare [118] observed that if one knows the speed of convergence to a given weak tangent, then one can get stronger lower bounds on dimension; that is, one can bound the Assouad spectrum, not just the Assouad dimension.…”

Section: Weak Tangents For Spectra?mentioning

confidence: 99%

“…However, one can always pass to a subsequence of the T k such that the convergence condition holds for all k, and so this yields no advantage. Several variants on Theorem 5.3.1 are considered in [118,Theorem 6]. For example, if one assumes the c k grow at most exponentially, then the result may be strengthened, and the similarity condition on T k may be weakened somewhat.…”

Section: Weak Tangents For Spectra?mentioning

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: Relationships Between Dimensionsmentioning

confidence: 94%

Section: Relationships Between Dimensionsmentioning

confidence: 94%

Section: Further Basic Propertiesmentioning

confidence: 99%

Section: Weak Tangents For Spectra?mentioning

confidence: 99%

Section: Weak Tangents For Spectra?mentioning

confidence: 99%

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Assouad Dimension and Fractal Geometry

Fraser

2020

111

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Generalised intermediate dimensions

Amlan

2023

Monatsh Math

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We introduce a family of dimensions, which we call the $$\Phi $$ Φ -intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. This is done by restricting the relative sizes of the covering sets in a way that allows for greater refinement than in the definition of the intermediate dimensions. We also extend the theory from Euclidean space to a wider class of metric spaces. We prove that these dimensions can be used to ‘recover the interpolation’ between the Hausdorff and box dimensions of compact subsets for which the intermediate dimensions are discontinuous at $$\theta =0$$ θ = 0 , thus providing finer geometric information about such sets. We prove continuity-like results involving the Assouad and lower dimensions, which give a sharp general lower bound for the intermediate dimensions that is positive for all $$\theta \in (0,1]$$ θ ∈ ( 0 , 1 ] for sets with positive box dimension. We also prove Hölder distortion estimates, a mass distribution principle, and a Frostman type lemma, which we use to study dimensions of product sets.

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Assouad-like dimensions of a class of random Moran measures

Hare

Mendivil

2022

Journal of Mathematical Analysis and Applications

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Properties of quasi-Assouad dimension

Cited by 6 publications

References 21 publications

Assouad Dimension and Fractal Geometry

Assouad Dimension and Fractal Geometry

Generalised intermediate dimensions

Assouad-like dimensions of a class of random Moran measures

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