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