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A fundamental characteristic of many mathematical spaces (e.g. vector spaces, metric spaces, topological spaces, etc.) is their dimension, which measures the “complexity” or “degrees of freedom” inherent in the space. There is no single notion of dimension; instead, there are a variety of different versions of this concept, with different versions being suitable for different classes of mathematical spaces. Typically, a single mathematical object may have several subtly different notions of dimension that one can place on it, which will be related to each other, and which will often agree with each other in “non-pathological” cases, but can also deviate from each other in many other situations. For instance:

• One can define the dimension of a space ${X}$ by seeing how it compares to some standard reference spaces, such as ${{\bf R}^n}$ or ${{\bf C}^n}$; one may view a space as having dimension ${n}$ if it can be (locally or globally) identified with a standard ${n}$-dimensional space. The dimension of a vector space or a manifold can be defined in this fashion.
• Another way to define dimension of a space ${X}$ is as the largest number of “independent” objects one can place inside that space; this can be used to give an alternate notion of dimension for a vector space, or of an algebraic variety, as well as the closely related notion of the transcendence degree of a field. The concept of VC dimension in machine learning also broadly falls into this category.
• One can also try to define dimension inductively, for instance declaring a space ${X}$ to be ${n}$-dimensional if it can be “separated” somehow by an ${n-1}$-dimensional object; thus an ${n}$-dimensional object will tend to have “maximal chains” of sub-objects of length ${n}$ (or ${n+1}$, depending on how one initialises the chain and how one defines length). This can give a notion of dimension for a topological space or a commutative ring.

The notions of dimension as defined above tend to necessarily take values in the natural numbers (or the cardinal numbers); there is no such space as ${{\bf R}^{\sqrt{2}}}$, for instance, nor can one talk about a basis consisting of ${\pi}$ linearly independent elements, or a chain of maximal ideals of length ${e}$. There is however a somewhat different approach to the concept of dimension which makes no distinction between integer and non-integer dimensions, and is suitable for studying “rough” sets such as fractals. The starting point is to observe that in the ${d}$-dimensional space ${{\bf R}^d}$, the volume ${V}$ of a ball of radius ${R}$ grows like ${R^d}$, thus giving the following heuristic relationship $\displaystyle \frac{\log V}{\log R} \approx d \ \ \ \ \ (1)$

between volume, scale, and dimension. Formalising this heuristic leads to a number of useful notions of dimension for subsets of ${{\bf R}^n}$ (or more generally, for metric spaces), including (upper and lower) Minkowski dimension (also known as box-packing dimension or Minkowski-Bougliand dimension), and Hausdorff dimension.

[In ${K}$-theory, it is also convenient to work with “virtual” vector spaces or vector bundles, such as formal differences of such spaces, and which may therefore have a negative dimension; but as far as I am aware there is no connection between this notion of dimension and the metric ones given here.]

Minkowski dimension can either be defined externally (relating the external volume of ${\delta}$-neighbourhoods of a set ${E}$ to the scale ${\delta}$) or internally (relating the internal ${\delta}$-entropy of ${E}$ to the scale). Hausdorff dimension is defined internally by first introducing the ${d}$-dimensional Hausdorff measure of a set ${E}$ for any parameter ${0 \leq d < \infty}$, which generalises the familiar notions of length, area, and volume to non-integer dimensions, or to rough sets, and is of interest in its own right. Hausdorff dimension has a lengthier definition than its Minkowski counterpart, but is more robust with respect to operations such as countable unions, and is generally accepted as the “standard" notion of dimension in metric spaces. We will compare these concepts against each other later in these notes.

One use of the notion of dimension is to create finer distinctions between various types of “small" subsets of spaces such as ${{\bf R}^n}$, beyond what can be achieved by the usual Lebesgue measure (or Baire category). For instance, a point, line, and plane in ${{\bf R}^3}$ all have zero measure with respect to three-dimensional Lebesgue measure (and are nowhere dense), but of course have different dimensions ( ${0}$, ${1}$, and ${2}$ respectively). (The Kakeya set conjecture, discussed recently on this blog, offers another good example.) This can be used to clarify the nature of various singularities, such as that arising from non-smooth solutions to PDE; a function which is non-smooth on a set of large Hausdorff dimension can be considered less smooth than one which is non-smooth on a set of small Hausdorff dimension, even if both are smooth almost everywhere. While many properties of the singular set of such a function are worth studying (e.g. their rectifiability), understanding their dimension is often an important starting point. The interplay between these types of concepts is the subject of geometric measure theory.

I’ve just uploaded a new paper to the arXiv entitled “A quantitative form of the Besicovitch projection theorem via multiscale analysis“, submitted to the Journal of the London Mathematical Society. In the spirit of my earlier posts on soft and hard analysis, this paper establishes a quantitative version of a well-known theorem in soft analysis, in this case the Besicovitch projection theorem. This theorem asserts that if a subset E of the plane has finite length (in the Hausdorff sense) and is purely unrectifiable (thus its intersection with any Lipschitz graph has zero length), then almost every linear projection E to a line will have zero measure. (In contrast, if E is a rectifiable set of positive length, then it is easy to show that all but at most one linear projection of E will have positive measure, basically thanks to the Rademacher differentiation theorem.)

A concrete special case of this theorem relates to the product Cantor set K, consisting of all points (x,y) in the unit square $[0,1]^2$ whose base 4 expansion consists only of 0s and 3s. This is a compact one-dimensional set of finite length, which is purely unrectifiable, and so Besicovitch’s theorem tells us that almost every projection of K has measure zero. (One consequence of this, first observed by Kahane, is that one can construct Kakeya sets in the plane of zero measure by connecting line segments between one Cantor set and another.)