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