Introduction

Approximation of real and complex numbers by rationals and algebraic numbers appeared first in papers by Dirichlet, Liouville and Hermite on Diophantine approximation and the theory of transcendental numbers. During the first three decades of the 20th century, E. Borel and A. Khintchine introduced the so-called metric (or measure theoretic) approach in which one considers approximation to any number which does not belong to an exceptional null set (i.e., a set of measure zero). Neglecting such exceptional sets can lead to strikingly simple and general theorems, such as Khintchine's theorem (see below). The exceptional sets can be analysed more deeply by using Hausdorff dimension, which can distinguish between different null sets.

This article gives an account of results, methods and ideas connected with Lebesgue measure and Hausdorff dimension of such exceptional sets. We will be concerned mainly with the lower bound of the Hausdorff dimension. Although determining the correct lower bound for the Hausdorff dimension of a set is often (though by no means always) harder than determining the correct upper bound, recent developments indicate that for many problems, the correct lower bound can be established using information associated with the upper bound. There are some exceptions to this principle. For example, convergence in the Khintchine–Groshev type theorem (for terminology see Bernik & Dodson 1999) for the parabola is related to the upper bound which was proved in Bernik (1979). Nevertheless the divergence case is still unsettled.

Introduction

Constructing rational approximants of numbers, such as π and the Metonic ratio, which occur in geometry and in the physical world, relies on Diophantine approximation of some sort. It is likely that Diophantine approximation was used to design sophisticated gearing 2500 years ago. Other applications arise from the natural link between rational dependence (corresponding to a Diophantine equation) and the physical phenomenon of resonance and so between Diophantine approximation and proximity to resonance. This can give rise to the ‘notorious problem of small denominators’1 in which solutions contain denominators that can become arbitrarily small, thus making convergence problematic. Small denominators occur in the study of the stability of the solar system (the N-body problem) and in dynamical systems; the fundamental character of these problems has inspired much new mathematics (fuller accounts are in). They also occur in related questions, such as averaging, and linearisation and normal forms.

The basic idea is to exclude sets of denominators which prevent convergence without significantly affecting the validity of the solution. Small denominators are related to very well approximable points, which lie in the complements of sets of points of certain Diophantine type; these complements include badly approximable numbers and their higher dimensional counterparts. The techniques developed in the metric theory of Diophantine approximation lend themselves to the study of the distribution of small denominators and in particular to the analysis of the Hausdorff dimension of the associated exceptional sets.

Introduction

Hausdorff measure and dimension stem from F. Hausdorff's simple but farreaching variation of C. Carathéodory's approach to Lebesgue measure (more details are given in the Notes at the end of the chapter). For familiar sets such as the interval, circle, sphere and the plane, the Hausdorff dimension (defined below in §3.3) coincides with the usual notion of dimension and is respectively 1,1,2 and 2. However, an important difference is that any set in Euclidean space has a Hausdorff dimension. In particular, null sets have a Hausdorff dimension and this gives a way of discriminating between them. The study of this finer aspect of the metric structure of exceptional sets, which started with Hausdorff's determination of the dimension of the Cantor ‘middle third’ set, was developed by A. S. Besicovitch and V. Jarník and continues unabated.

Hausdorff measure has been studied intensively and in considerable generality, indeed the theory can be extended to a metric space setting. This tract will be concerned mainly with Borel subsets of submanifolds of Euclidean space and accordingly the treatment of Hausdorff measure and dimension will be in ℝn. Fuller treatments and further references can be found in the books of K. Falconer, H. Federer, P. Mattila and C. A. Rogers. Applications to exceptional sets in number theory are discussed in.

Hausdorff measure

Hausdorff measure is based on covers. Let E be a set in ℝn and let s be a non-negative real number.

This book is about metric Diophantine approximation on smooth manifolds embedded in Euclidean space. The aim is to develop a coherent body of theory on the lines of that which already exists for the classical theory, corresponding to the manifold being Euclidean space. Although the functional dependence of the coordinates presents serious technical difficulties, there is a surprising degree of interplay between the very different areas of number theory, differential geometry and measure theory.

A systematic theory began to emerge in the mid–1960's when V. G. Sprindžuk and W. M. Schmidt established that certain types of curve were extremal (an extremal set enjoys the property that, in a sense that can be made precise, Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved for almost all points in the set; thus the real line is extremal). Sprindžuk conjectured that analytic manifolds satisfying a necessary nondegeneracy condition are extremal. Over the last 30 years, there has been considerable progress in verifying this conjecture for manifolds satisfying various arithmetic and geometric constraints, culminating in its recent proof by D. Y. Kleinbock and G. A. Margulis using ideas of flows on homogeneous spaces of lattices. The greater part of this book is concerned with establishing the counterparts of Khintchine's theorem for manifolds and with the Hausdorff dimension of the associated exceptional sets.