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.