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Preface
Acknowledgements
Theory
Fractal Geometry and Dimension Theory
The Emergence of Fractal Geometry
Dimension Theory
The Assouad Dimension
The Assouad Dimension and a Simple Example
A Word or Two on the Definition
Some History
Basic Properties: The Greatest of All Dimensions
Some Variations on the Assouad Dimension
The Lower Dimension
The Quasi-Assouad Dimension
The Assouad Spectrum
Basic Properties: Revisited
Dimensions of Measures
Assouad and Lower Dimensions of Measures
Assouad Spectrum and Box Dimensions of Measures
Weak Tangents and Microsets
Weak Tangents and the Assouad Dimension
Weak Tangents for the Lower Dimension?
Weak Tangents for Spectra?
Weak Tangents for Measures?
Examples
Iterated Function Systems
IFS Attractors and Symbolic Representation
Invariant Measures
Dimensions of IFS Attractors
Ahlfors Regularity and Quasi-Self-Similarity
Self-Similar Sets
Self-Similar Sets and the Hutchinson–Moran Formula
The Assouad Dimension of Self-Similar Sets
The Assouad Spectrum of Self-Similar Sets
Dimensions of Self-Similar Measures
Self-Affine Sets
Self-Affine Sets and Two Strands of Research
Falconer’s Formula and the Affinity Dimension
Self-Affine Carpets
Self-Affine Sets with a Comb Structure
A Family of Worked Examples
Dimensions of Self-Affine Measures
Further Examples: Attractors and Limit Sets
Self-Conformal Sets
Invariant Sets for Parabolic Interval Maps
Limit Sets of Kleinian Groups
Mandelbrot Percolation
Geometric Constructions
Products
Orthogonal Projections
Slices and Intersections
Two Famous Problems in Geometric Measure Theory
Distance Sets
Kakeya Sets
Conformal Dimension
Lowering the Assouad Dimension by Quasi-Symmetry
Applications
Applications in Embedding Theory
Assouad’s Embedding Theorem
The Spiral Winding Problem
Almost Bi-Lipschitz Embeddings
Applications in Number Theory
Arithmetic Progressions
Diophantine Approximation
Definability of the Integers
Applications in Probability Theory
Uniform Dimension Results for Fractional Brownian Motion
Dimensions of Random Graphs
Applications in Functional Analysis
Hardy Inequalities
Lp → Lq Bounds for Spherical Maximal Operators
Connection with Lp-Norms
Future Directions
Finite Stability of Modified Lower Dimension
Dimensions of Measures
Weak Tangents
Further Questions of Measurability
IFS Attractors
Random Sets
General Behaviour of the Assouad Spectrum
Projections
Distance Sets
The Holder Mapping Problem and Dimension
Dimensions of Graphs
References
List of Notation
Index