The examples we saw in Chapters 1 and 2 suggest that real numbers are arithmetically quite diverse. The theory of continued fractions as we have developed it allows us to recognise whether a given real number is rational or is a quadratic irrational; for the latter as well as for several transcendental numbers such as e, whose quotients follow a clear periodic pattern, we have precise knowledge of the quality of their rational approximations, as for instance in (2.40).

A standard counting argument, however, shows that the totality of such numbers is countable; hence they form a subset of measure zero of the reals. It is therefore reasonable to look into the arithmetic properties of other real numbers – in particular, of almost all real numbers (of course, in the sense of the usual Lebesgue measure M).

The classical problems of metric number theory include determining the measure of the set of numbers that satisfy a given arithmetic property. In the context of continued fractions, for example, we may ask about the measure of the set of numbers whose 100th quotient a100 is exactly 100, or whose 100th convergent pn/qn satisfies qn < 1010. This is exactly the sort of question that we will address in this chapter.