Glossary of terms

We list a few notations, definitions and statements used in the main text. For what still remains unexplained or unproved we refer to the usual textbooks on elementary analysis, topology and algebra.

Sets

Let Y, Z be sets. If Z ⊂ Y then Y\Z: = {y ∈ Y : y ∉ Z}. The product set Y × Z equals {(y, z) : y ∈ Y, z ∈ Z}. We write Y2 : = Y × Y, Yn+1 : = Yn × Y (n ∈ ℕ, n ≥ 2). Y is countable if there exists a surjection ℕ → Y, otherwise Y is uncountable. The cardinality of Y is strictly less than the cardinality of Z if no map Y → Z is surjective. A partition of Y is a covering of Y by mutually disjoint subsets. The classes of an equivalence relation ∼ on Y form a partition of Y. The set of these classes is denoted Y/∼. The quotient map π : Y → Y/∼ sends each element x ∈ Y into its class π(x). A (full) set of representatives in Y of ∼, or a (full) set of representatives in Y modulo ∼ is a subset R of Y such that π maps R bijectively onto Y/∼.

Let Y = (Y, ≥) be a partially ordered set. A maximal element of Y is an element y ∈ Y such that z ∈ Y, z ≥ y implies z = y.

The geometric measure theory of sets of integral and fractional dimension has been developed by pure mathematicians from early in this century. Recently there has been a meteoric increase in the importance of fractal sets in the sciences. Mandelbrot (1975,1977,1982) pioneered their use to model a wide variety of scientific phenomena from the molecular to the astronomical, for example: the Brownian motion of particles, turbulence in fluids, the growth of plants, geographical coastlines and surfaces, the distribution of galaxies in the universe, and even fluctuations of price on the stock exchange. Sets of fractional dimension also occur in diverse branches of pure mathematics such as the theory of numbers and non-linear differential equations. Many further examples are described in the scientific, philosophical and pictorial essays of Mandelbrot. Thus what originated as a concept in pure mathematics has found many applications in the sciences. These in turn are a fruitful source of further problems for the mathematician. This tract is concerned primarily with the geometric theory of such sets rather than with applications.

The word ‘fractal’ was derived from the latin fractus, meaning broken, by Mandelbrot (1975), who gave a ‘tentative definition’ of a fractal as a set with its Hausdorff dimension strictly greater than its topological dimension, but he pointed out that the definition is unsatisfactory as it excludes certain highly irregular sets which clearly ought to be thought of in the spirit of fractals. Hitherto mathematicians had referred to such sets in a variety of ways – ‘sets of fractional dimension’, ‘sets of Hausdorff measure’, ‘sets with a fine structure’ or ‘irregular sets’.

Introduction

In this chapter we discuss the density and tangency structure of s-sets in ℝn when s is an integer. We know from Corollary 2.10 that an s-set splits into a regular part and an irregular part, and we find that these two types of set exhibit markedly different properties. One of our aims is to characterize regular sets as subsets of countable unions of rectifiable curves or surfaces, and thus to relate the measure theoretic and the descriptive topological ideas.

We present in detail the theory of linearly measurable sets or 1-sets in ℝ2. This work is almost entirely due to Besicovitch (1928a, 1938), the latter paper including some improved proofs as well as further results. Most of his proofs seem hard to better except in relatively minor ways and, hopefully, in presentation. Certainly, some of the geometrical methods used by Besicovitch involve such a degree of ingenuity that it is surprising that they were ever thought of at all. Some of the work in this chapter is also described in de Guzman (1981).

Curves and continua

Regular 1-sets and rectifiable curves are intimately related. Indeed, a regular 1-set is, to within a set of measure zero, a subset of a countable collection of rectifiable curves. This section is devoted to a study of curves, mainly from a topological viewpoint and in relation to continua of finite linear measure. Here we work in ℝn as the theory is no more complicated than for plane curves.

A curve (or Jordan curve) Γ is the image of a continuous injection ψ:[a,b]→ℝn, where [a,b]⊂:ℝ is a closed interval. Any curve is a continuum, that is, a compact connected set.

Introduction

The Kakeya problem has an interesting history. In 1917 Besicovitch was working on problems on Riemann integration, and was confronted with the following question: if f is a Riemann integrable function defined on the plane, is it always possible to find a pair of orthogonal coordinate axes with respect to which ∫ f(x,y)dx exists as a Riemann integral for all y, and with the resulting function of y also Riemann integrable? Besicovitch noticed that if he could construct a compact set F of plane Lebesgue measure zero containing a line segment in every direction, this would lead to a counter-example: For assume, by translating F if necessary, that F contains no segment parallel to and rational distance from either of a fixed pair of axes. Let f be the characteristic function of the set F0 consisting of those points of F with at least one rational coordinate. As F contains a segment in every direction on which both F0 and its complement are dense, there is a segment in each direction on which f is not Riemann integrable. On the other hand, the set of points of discontinuity of F is of plane measure zero, so f is Riemann integrable over the plane by the wellknown criterion of Lebesgue.

Besicovitch (1919) succeeded in constructing a set, known as a ‘Besicovitch set’, with the required properties. Owing to the unstable situation in Russia at the time, his paper received limited circulation, and the construction was later republished in Mathematische Zeitschrift (1928).