1. Introduction 1.1. Background: the basic example. To set the scene for the abstract framework considered in this article we introduce a basic limsup set whose study has played a central role in the development of the classical theory of metric Dio- phantine approximation. Given a real, positive decreasing function ip : M+ M+, let W(I/J) := {x G [0,1] : \x p/q\ ip(q) for i.m. rationals p/q (q 0)}, where 'i.m.' means 'infinitely many'. This is the classical set of -0-well approximable numbers in the theory of one dimensional Diophantine approximation. The fact that we have restricted our attention to the unit interval rather than the real line is purely for convenience. It is natural to refer to the function ip as the approximating function. It governs the 'rate' at which points in the unit interval must be approximated by rationals in order to lie in W{ijj). It is not difficult to see that W(tp) is a limsup set. For n e N, let W(^n):= |J |J B(p/q^(q))n[0A} kri~1qkn 0pq where k 1 is fixed and B(c,r) is the open interval centred at c of radius r. The set W(ip) consists precisely of points in the unit interval that lie in infinitely many W(ij))ri)', that is oo oo W{il) = limsupTy(^,n) := f] ( J W(ip,n) . m= l n=ra Investigating the measure theoretic properties of the set W(I/J) underpins the classical theory of metric Diophantine approximation. We begin by considering the 'size' of W(ip) expressed in terms of the ambient measure m i.e. one-dimensional Lebesgue measure. On exploiting the limsup nature of W(tp), a straightforward application of the convergence part of the Borel-Cantelli lemma from probability theory yields that oo m(W(il)) = 0 if ]T/c 2 (/c n ) oo . 7 1 = 1 Notice that since ip is monotonic, the convergence/divergence property of the above sum is equivalent to that of Yl^Li r ^{r)- l
Previous Page Next Page