Introduction

This chapter examines local properties of s-sets in ℝn for nonintegral s. The fundamental result is that any such set is irregular, that is, has lower circular or spherical density strictly less than 1 at almost all of its points. Indeed, the stronger result that its density fails to exist at almost all of its points has also been established. As before, we also examine the existence of suitably defined tangents, and show that the set of points at which such tangents exist must have measure zero.

For the case of subsets of the plane, the work is entirely due to Marstrand (1954a, 1955), the former paper providing a very complete account. As with sets of integral dimension, higher-dimensional analogues present formidable difficulties; the natural generalizations were eventually proved by Marstrand (1964).

s-sets with 0 < s < 1

First we consider s-sets in ℝn for s strictly less than 1. In this case the basic properties, including non-existence of the density almost everywhere, are relatively easy to obtain.

The following topological observation about such sets is sometimes useful.

Lemma 4.1

An s-set E in ℝnwith 0 < 5 < 1 is totally disconnected.

Proof. Let x and y be distinct points in the same connected component of E. Define a mapping f:ℝn →[0, ∞) by f(z) = |z–x|. Since f does not increase distances it follows from Lemma 1.8 that ℋs(f(E)) ≤ ℋs(E) < ∞. As 5 < 1 it follows that f(E) is a subset of ℝ of Lebesgue measure zero and, in particular, has dense complement.

From now on we shall use the following conventions. Unless stated explicitly otherwise we have

K IS A COMPLETE NON-ARCHIMEDEAN NON-TRIVIALLY VALUED FIELD WITH RESIDUE CLASS FIELD k

THE CHARACTERISTIC OF A FIELD L IS char (L)

p IS A PRIME NUMBER

x = Σnanpn FOR A p-ADIC NUMBER x DENOTES THE STANDARD EXPANSION OF x (see Section 5)

IF char(K) = 0, char(k) = p THEN THE VALUATION IS CHOSEN SUCH THAT ℝp IS A VALUED SUBFIELD OF K. THE VALUATION ON ℂp IS DENOTED │ │p

LET K HAVE A DISCRETE VALUATION. THEN π ∈ K IS AN ELEMENT SUCH THAT │φ│ = max │K×│ ∩ (0, 1)

PART 1: ELEMENTARY CALCULUS

In Chapter 2 we shall develop the first principles of ultrametric calculus. Our main interest lies in calculus over ℚp and ℂp.

The classical concepts of calculus

This section is not very exciting. We shall list notions and statements that are directly borrowed from the classical analysis over ℝ and ℂ, some of which have been used already implicitly in Chapter 1. Starting with the next section the story of ultrametric calculus is going to diverge from the ‘classical’ one and shall become more interesting. No proofs are given; in case of any doubt the reader may supply one as an exercise.

Recall that our assumption that K is complete means that every Cauchy sequence in K converges. A sequence a1, a2, … in K converges to an element a ∈ K if limn → ∞ │a − an│ = 0.

This tract provides a rigorous self-contained account of the mathematics of sets of fractional and integral Hausdorff dimension. It is primarily concerned with geometric theory rather than with applications. Much of the contents could hitherto be found only in original mathematical papers, many of which are highly technical and confusing and use archaic notation. In writing this book I hope to make this material more readily accessible and also to provide a useful and precise account for those using fractal sets.

Whilst the book is written primarily for the pure mathematician, I hope that it will be of use to several kinds of more or less sophisticated and demanding reader. At the most basic level, the book may be used as a reference by those meeting fractals in other mathematical or scientific disciplines. The main theorems and corollaries, read in conjunction with the basic definitions, give precise statements of properties that have been rigorously established.

To get a broad overview of the subject, or perhaps for a first reading, it would be possible to follow the basic commentary together with the statements of the results but to omit the detailed proofs. The non-specialist mathematician might also omit the details of Section 1.1 which establishes the properties of general measures from a technical viewpoint.

A full appreciation of the details requires a working knowledge of elementary mathematical analysis and general topology. There is no doubt that some of the proofs central to the development are hard and quite lengthy, but it is well worth mastering them in order to obtain a full insight into the beauty and ingenuity of the mathematics involved.

Introduction

This chapter surveys examples of sets of fractional dimension which result from particular constructions or occur in other branches of mathematics or physics and relates them to earlier parts of the book. The topics have been chosen very much at the author's whim rather than because they represent the most important occurrences of fractal sets. In each section selected results of interest are proved and others are cited. It is hoped that this approach will encourage the reader to follow up some of these topics in greater depth elsewhere. Most of the examples come from areas of mathematics which have a vast literature; therefore in this chapter references are given only to the principal sources and to recent papers and books which contain further surveys and references.

Curves of fractional dimension

In this section we work in the (x,y)-coordinate plane and investigate the Hausdorff dimension of Γ, the set of points (x,f(x)) forming the graph of a function f defined, say, on the unit interval.

If f is a function of bounded variation, that is, if is bounded for all dissections 0 = x0 < x1 < … < xm = l, then we are effectively back in the situation of Section 3.2; Γ is a rectifiable curve and so a regular 1-set. However, if f is a sufficiently irregular, though continuous, function it is possible for Γ to have dimension greater than 1. In such cases it can be hard to calculate the Hausdorff dimension and measure of Γ from a knowledge of f. However, if f satisfies a Lipschitz condition it is easy to obtain an upper bound.

This elementary book is intended for advanced undergraduates or anyone on a higher level who wants to learn the basic facts of p-adic analysis. We only assume the reader to have some standard knowledge of analysis and algebra.

In analysis (and outside it) the fields ℝ and ℂ play a central role. For several reasons people started to study the implications of replacing ℝ or ℂ by a more general object, viz. a field K with a complete valuation │ │ comparable to the absolute value function (see Definition 1.1). Many such fields other than ℝ or ℂ exist, their valuations are all ‘non-archimedean’, i.e. they satisfy the ‘strong triangle inequality’ │x + y│ ≤ max(│x│, │y│). The analysis in and over non-archimedean valued fields K is known as ultrametric (non-archimedean, p-adic) analysis.

In this book we shall treat the basic facts of ultrametric analysis together to form an alternative ‘one variable calculus course’. Thus, in K we shall consider familiar concepts such as continuity, differentiability, (power) series, integration, etc. However, the strong triangle inequality causes fascinating deviations from the ‘classical analysis’ (over ℝ or ℂ); let us mention a few of them.

1. (i) A series Σan in K converges if limn → ∞an = 0. The power series Σxn/n! of exp (if it makes sense at all) converges only on a disc strictly contained in the closed unit disc {x : │x│ ≤ 1}. Hence Σ1/n! diverges (but Σn! converges in many K).