1. INTRODUCTION 3
JARNIK'S THEOREM
(1931). Let f be a dimension function such
thatr~x
f(r)
oo as r —- 0 and r _ 1 f(r) is decreasing. Let ^ be a real, positive decreasing function.
Then
,' 0
if
E~i r f Mr)) oo
0 0 if YZLi r f (i/;(r)) = 00
Clearly the above theorem can be regarded as the Hausdorff measure version
of Khintchine's theorem. As with the latter, the divergence part constitutes the
main substance. Notice, that the case when H* is comparable to one-dimensional
Lebesgue measure m (i.e. f(r) = r) is excluded by the condition r _ 1 f(r) 00 as
r 0. Analogous to Khintchine's original statement, in Jarnik's original statement
the additional hypotheses that r2ip(r) is decreasing, r2,ip(r) » 0 as r 00 and
that
r2f(ip(r))
is decreasing were assumed. Thus, even in the simple case when
f(r) = rs (s 0) and the approximating function is given by i/j(r) = r _ r l o g r
(r 2), Jarnik's original statement gives no information regarding the s-dimensional
Hausdorff measure of W(ip) at the critical exponent s = 2/r - see below. That
this is the case is due to the fact that
r2f(t/j(r))
is not decreasing. However, as we
shall see these additional hypotheses are unnecessary. More to the point, Jarnik's
theorem as stated above is the precise Hausdorff measure version of Khintchine's
theorem. Of course, as with Khintchine's theorem the question of removing the
monotonicity condition on the approximating function tp now arises. That is to say,
it now makes perfect sense to consider a generalized Duffin-Schaeffer conjecture for
Hausdorff measures - for a detailed account regarding the original Duffin-Schaeffer
conjecture see [22, 39]. Briefly, let ijj(n) be a sequence of non-negative real numbers
and consider the set W(ip) of x £ [0,1] for which there exist infinitely many rationals
P/Q ( 9 ^ 1 ) such that
\x p/q\ ip(q) with (p, q) = 1 .
The Duffin-Schaeffer conjecture for Hausdorff measures: Let / be a dimension func-
tion such that
r~l
f(r) 00 as r 0 and r
- 1
f(r) is decreasing. Let (p denote the
Euler function. Then
00
Hf
(WW*)) - 00 if ] T / W(n)) (j){n) = 00 .
n= l
It is easy to show that 7i^(W(ip)) = 0 if the above sum converges. The higher
dimensional Duffin-Schaeffer conjecture corresponding to simultaneous approximation
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