# Dimensions & Embeddings in Analysis

This course will investigate four different definitions of "the dimension" and show that they can be understood in terms of how "finite-dimensional" sets can be embedded into Euclidean spaces (i.e. the existence of a one-to-one mapping with certain well-defined continuity properties).

The fundamental question is how to embed metric spaces for which the dimension (whichever one we are considering) is finite, but one can reduce this question to subsets of Banach spaces and (with some loss) subsets of Hilbert spaces.

For a rough overview, we will look at:

- The Lebesgue covering dimension (from topology) and embeddings using continuous maps

- The Hausdorff dimension, the most classical non-integer dimension, much used in geometric measure theory and dynamical systems and embeddings of subsets of Banach spaces using linear maps

These first two embedding results rely on the Baire Category Theorem

- The box-counting dimension, for which perhaps the sharpest results are available: embeddings of subsets of Hilbert/Banach spaces via linear maps with Holder continuous inverses

- The Assouad dimension, which arises naturally in the study of the existence of bi-Lipschitz embeddings of metric spaces into Euclidean spaces. This question is still open, but I will give Assouad's proof of the bi-Lipschitz embedding of the spaces $(X,d^\alpha)$ for any $0<\alpha<1$ and present some results on "almost bi-Lipschitz" embeddings of subsets of Hilbert spaces that are closely related.

Here are the notes for lectures. Please let me know of any typos.

The main reference is

J.C. Robinson (2011) *Dimensions, Embeddings, & Attractors*. Cambridge Tracts in Mathematics 186. Cambridge University Press.

but some of the material will be presented in alternative ways, and some additional material included.

If you would like an assessment, then here are some exercises. You should do at least five of the eight.

**Lecture One - Lebesgue covering dimension and the fundamental embedding theorem**

Definition of the Lebesgue covering dimension; proof that for any compact metric space $(X,d)$ with dim$(X)=n$, a residual set in $C(X;{\mathbb R}^{2d+1})$ provide embeddings of $X$ into ${\mathbb R}^{2d+1}$.

W. Hurewicz & H. Wallman (1941) *Dimension Theory. *Princeton University Press.

J.R. Munkres (2000) *Topology. *2nd edition. Prentice Hall, Upper Saddle River, NJ.

**Lecture Two - Hausdorff dimension and its relation to covering dimension**

Definition and properties of the Hausdorff dimension. Proof that dim$(X)\le{\rm dim}_{\rm H}(X)$.

K.J. Falconer (1985) *The Geometry of Fractal Sets. *Cambridge Tracts in Mathematics 85. Cambridge University Press.

G.A. Edgar (2008) *Measure, Topology, and Fractal Geometry. *Undergraduate Texts in Mathematics. Springer, New York.

**Lecture Three - Isometric embeddings of metric spaces into Banach spaces and linear embeddings into Euclidean spaces**

Any compact metric space $(X,d)$ can be isometrically embedded into $L^\infty(X)$ and into $\ell^\infty$. If $X$ is a compact subset of a real Banach space $B$ with ${\rm dim}_{\rm H}(X-X)<d$ then a residual set of maps in $L(B;{\mathbb R}^{d+1})$ provide embeddings of $X$ into ${\mathbb R}^{d+1}$.

A. Ben-Artzi, A. Eden, C. Foias, & B. Nicolaenko (1993) Holder continuity for the inverse of Mane's projection. *J. Math. Anal. Appl. ***178**, 22-29.

J. Heinonen (2001) *Geometric embeddings of metric spaces. *Report Univeristy of Jyvaskyla Department of Mathematics and Statistics 90.

R. Mané (1981) On the dimension of the compact inavriant sets of certain nonlinear maps. Springer LNM 898, 230-242.

**Lecture Four - The box-counting dimension: properties and examples**

Definition of the box-counting dimension in terms of coverings by open balls. Proof of various properties, including ${\rm dim}_{\rm H}(X)\le{\rm dim}_{\rm B}(X)$. Some examples, including orthogonal sequences in $\ell^p$ spaces. Embedding subsets of Banach spaces with ${\rm dim}_{\rm B}(X)<\infty$ into Hilbert spaces with Holder continuous inverse.

Some material is new; the rest is fairly standard and can be found in my book, or in

K.J. Falconer (1990) *Fractal geometry.* Wiley, Chichester*. *

The formula for the dimension of "orthogonal" sequences in a Hilbert space is due to Ben-Artzi et al. (see paper above); the extension to $\ell^p$ spaces is immediate.

**Lecture Five - A first embedding result for sets with finite box-counting dimension**

We show that if $X\subset{\mathbb R}^N$ and $N>k>2{\rm dim}_{\rm B}(X)$ then almost every linear map from ${\mathbb R}^N$ into ${\mathbb R}^k$ is injective on $X$ with H\"older continuous inverse.

We also show how to define a measure on a collection of linear maps from an infinite-dimensional Hilbert space $H$ into ${\mathbb R}^k$, given a sequence of linear subspaces $\{V_j\}$ of $H$; this construction will be used in the next lecture to prove a similar embedding result when $X\subset H$.

The proof of the embedding result for subsets of ${\mathbb R}^N$ is a finite-dimensional version of that developed by Hunt & Kaloshin.

B.R. Hunt & V.Y. Kaloshin (1999) Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. *Nonlinearity ***12, **1263-1275.

**Lecture Six - Embedding subsets of Hilbert spaces into ${\mathbb R}^k$**

We prove Hunt & Kaloshin's embedding result for subsets of a Hilbert space with finite box-counting dimension, with a Holder exponent depending on the box-counting dimension and the "thickness exponent" $\tau(X)$, which satisfies $\tau(X)\le{\rm dim}_{\rm B}(X)$.

As a corollary we obtain embedding results for metric spaces and subsets of Banach spaces with finite box-counting dimension: the embeddings are Lipschitz/linear one way and Holder continuous the other.

We also show, following Pinto de Moura & Robinson (2010) that the Holder exponent is sharp in the limit as the embedding dimension $k$ tends to infinity.

E. Pinto de Moura & J.C. Robinson (2010) Orthogonal sequences and regularity of embeddings into finite-dimensional spaces. *J. Math. Anal. Appl. ***368**, 254-262.

**Lecture Seven - Assouad's bi-Lipschitz embeddings of 'snowflaked' metric spaces $(X,d^\alpha)$**

We will define the Assouad dimension of a metric space, and prove an embedding theorem due to Assouad that for any metric space with ${\rm dim}_{\rm A}(X)<\infty$, the "snowflaked" spaces $(X,d^\alpha)$ admit bi-Lipschitz embeddings into Euclidean spaces for all $0<\alpha<1$. We will briefly discuss results on "almost bi-Lipschitz embeddings" due to Olson & Robinson.

P. Assouad (1983) Plongements lipschitziens dans ${\mathbb R}^n$. *Bull. Soc. Math. France 111,* 429-448.

E.J. Olson & J.C. Robinson (2010) Almost bi-Lipschitz embeddings and almost homogeneous sets. *Trans. Amer. Math. Soc. ***362**, 145-168.

J.C. Robinson (2009) Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces. *Nonlinearity ***22**, 711-728.

The missing "lecture eight" would have gone into the proof of the almost bi-Lipschitz embeddings in more detail. This can be found in Chapter 9 of Robinson (2011), although with an example showing that finite Assouad dimension is not sufficient to ensure the existence of a bi-Lipschitz embedding of a (compact) metric space into a Euclidean space.

It remains open whether finite Assouad dimension allows subsets of a Hilbert space to be bi-Lipschitz embedded into Euclidean sapces.