The box dimensions of exceptional self-affine sets in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>

Fraser, Jonathan M.; Jurga, Natalia

doi:10.1016/j.aim.2021.107734

Cited by 2 publications

(7 citation statements)

References 18 publications

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“…The simplicity of this model facilitated exact calculation of the dimensions, whilst maintaining many of the interesting features of self-affine sets. This strategy led to various different classes of selfaffine carpet being introduced with increasing levels of generality, see [122,15,83,86,172,168,102]. Traditionally, a 'carpet' is a planar selfaffine set generated by diagonal matrices.…”

Section: Self-affine Sets and Two Strands Of Researchmentioning

confidence: 99%

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Assouad Dimension and Fractal Geometry

Fraser

2020

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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: Self-affine Sets and Two Strands Of Researchmentioning

confidence: 99%

“…For example, the sets considered in [86] also allow anti-diagonal matrices and the examples in [172] allow triangular matrices. Also, higher dimensional analogues have been considered, where the term 'carpet' is often replaced by 'sponge' [168,49,102].…”

Section: Self-affine Sets and Two Strands Of Researchmentioning

confidence: 99%

Assouad Dimension and Fractal Geometry

Fraser

2020

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111

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“…The Lalley–Gatzouras class in arbitrary dimensions was also handled independently from our work in [32]. Recently, Fraser and Jurga [24] considered sponges in

d=3

in the more general setting where each diagonal matrix can be composed with a permutation matrix. Amongst sponges which satisfy the SPPC, their main result only covers the Lalley–Gatzouras class.…”

Section: Discussion and Two Worked Out Examplesmentioning

confidence: 99%

“…Amongst sponges which satisfy the SPPC, their main result only covers the Lalley–Gatzouras class. More importantly, they present an example in [24, Theorem 5.5] which shows that their bounds are not applicable in general to the Barański class. In Section 4.2, we calculate the box dimension of this sponge and show the qualitative difference of our pressure compared to the one in [24].…”

Section: Discussion and Two Worked Out Examplesmentioning

confidence: 99%

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The Lq$L^q$ spectrum of self‐affine measures on sponges

Kolossváry

2023

Journal of London Math Soc

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In this paper, a sponge in double-struckRd$\mathbb {R}^d$ is the attractor of an iterated function system consisting of finitely many strictly contracting affine maps whose linear part is a diagonal matrix. A suitable separation condition is introduced under which a variational formula is proved for the Lq$L^q$ spectrum of any self‐affine measure defined on a sponge for all q∈double-struckR$q\in \mathbb {R}$. Apart from some special cases, even the existence of their box dimension was not proved before. Under certain conditions, the formula has a closed form which in general is an upper bound. The Frostman and box dimension of these measures is also determined. The approach unifies several existing results and extends them to arbitrary dimensions. The key ingredient is the introduction of a novel pressure function which aims to capture the growth rate of box counting quantities on sponges. We show that this pressure satisfies a variational principle which resembles the Ledrappier–Young formula for Hausdorff dimension.

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The box dimensions of exceptional self-affine sets in R3

Cited by 2 publications

References 18 publications

Assouad Dimension and Fractal Geometry

Assouad Dimension and Fractal Geometry

The Lq$L^q$ spectrum of self‐affine measures on sponges

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