4 Beresnevich, Dickinson & Velani
is known to be true [37]. It is plausible that the ideas in [37] together with those in this
paper are sufficient to prove the higher dimensional version of the above conjecture.
The first and last authors have shown that this is indeed the case [9].
Returning to Jarnik's theorem, note that in the case when H,f is the standard
s-dimensional Hausdorff measure
Hs
(i.e. f(r) = r
s
), it follows from the definition
of Hausdorff dimension (see §7) that
dim W(il) = inf{s : £ £ i
rip(r)s
oo} .
Previously, Jarnik (1929) and independently Besicovitch (1934) had determined
the Hausdorff dimension of the set W(r i— r
_ r
) , usually denoted by W(r), of r-well
approximable numbers. They proved that for r 2, dimVF(r) = 2/r. Thus, as
the 'rate' of approximation increases (i.e. as r increases) the 'size' of the set W(r)
expressed in terms of Hausdorff dimension decreases. As discussed earlier, this is in
precise keeping with one's intuition. Obviously, the dimension result implies that
«• or™-/° " * ' ' ' .
[ oo if s 2/r
but gives no information regarding the s-dimensional Hausdorff measure of W(r)
at the critical value s = dimW(r). Clearly, Jarnik's zero-infinity law implies the
dimension result and that for r 2
H2/r(W(r)) = oo .
Furthermore, the 'zero-infinity' law allows us to discriminate between sets with the
same dimension and even the same s-dimensional Hausdorff measure. For example,
with r 2 and 0 ei €2 consider the approximating functions
^c.(r) := r~T (log
r
) " ^ ( 1 + l ) (i = 1,2) .
It is easily verified that for any ei 0,
™(W($ei))=Q, dimWrft/0 = 2/r and H2/r{W(ipei)) = 0 .
However, consider the dimension function / given by f(r) = r
2
/
r
(logr-
1
/
r
)
C l
. Then
Yl^Li r f (^(0) x
1L/7LI
(r (logr) 1 + i _ e i ) _ 1 , where as usual the symbol x denotes
comparability (the quotient of the associated quantities is bounded from above and
below by positive, finite constants). Hence, Jarnik's zero-infinity law implies that
Hf{W{ipei))
= 00 whilst
Hf
(W(rle2)) - 0.
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