2
Beresnevich, Dickinson & Velani
A natural problem now arises. Under what conditions is m(W(ip)) 0 ? The
following fundamental result provides a beautiful and simple criteria for the 'size' of
the set W(ip) expressed in terms of Lebesgue measure.
If ijj(r) is decreasing then
if
J2T=i rip(r) oo ,
if
E^Li r^{r) = oo .
Thus, in the divergence case, which constitutes the main substance of Khintchine's
theorem, not only do we have positive Lebesgue measure but full Lebesgue measure.
In fact, this turns out to be the case for all the examples considered in this paper.
Usually, there is a standard argument which allows one to deduce full measure from
positive measure - such as the invariance of the limsup set or some related set, under
an ergodic transformation. In any case, we shall prove a general result which directly
implies the above full measure statement. It is worth mentioning that in Khintchine's
original statement the stronger hypothesis that
r2ip(r)
is decreasing was assumed.
Returning to the convergence case, we cannot obtain any further information
regarding the 'size' of
W(I/J)
in terms of Lebesgue measure it is always zero. Intu-
itively, the 'size' of W(ip) should decrease as the rate of approximation governed by
the function ty increases. In short, we require a more delicate notion of 'size' than
simply Lebesgue measure. The appropriate notion of 'size' best suited for describ-
ing the finer measure theoretic structures of W(ip) is that of generalized Hausdorff
measures. The Hausdorff /-measure H- with respect to a dimension function / is
a natural generalization of Lebesgue measure. So as not to interrupt the flow of
this background/motivation exposition we referee the reader to §7 for the standard
definition of H^ and further comments regarding Hausdorff measures and dimension.
Again on exploiting the limsup nature of W(^), a straightforward covering ar-
gument provides a simple convergence condition under which 7i^(W(t/j)) = 0. Thus,
in view of the development of the Lebesgue theory it is natural to ask for conditions
under which H?(W(ip)) is strictly positive.
The following fundamental result provides a beautiful and simple criteria for the
'size' of the set W(ip) expressed in terms of Hausdorff measures.
KHINTCHINE'S THEOREM
(1924) .
0
m{W(il)))
1
Previous Page Next Page