Assouad Dimension and Fractal Geometry

Abstract

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...

Assouad dimension of planar self-affine sets

“…In the following example, we show that, under the assumptions of Theorem 3.2, it is possible to have dim L (X) < dim H (X) = dim aff (X) < dim A (X). This observation answers one of the open questions posed in [28,Question 17.5.2]. The example strongly relies on known results on Bedford-McMullen carpets and it is currently the sole example to demonstrate dim aff (X) < dim A (X) in the case dim H (X) 1.…”

“…The important observation here is that if the lower and Assouad dimensions of X differ, then X cannot be Ahlfors regular. For other basic properties of the Assouad and lower dimensions, we refer to the book of Fraser [28].…”

Intermediate dimensions

“…Assouad dimension has been studied intensively in recent years, see the books [13,24] and paper [11].…”

“…If s = m this reduces to 13) which are the kernels φ m r (x) used in the context of box-counting dimensions [6,7]. Note that φ s,m r,θ (x) is continuous in x and monotonically decreasing in |x|.…”

Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

“…The Assouad dimension was introduced by Assouad [2,3] in relation to quasiconformal mappings and embeddability problems, see [10,13,17]. Recently though, considerable attention has been given to its study in Fractal geometry, see [1,5,6,8,9,11,14] or the recent book [7] to name a few. The Assouad dimension of a set…”