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.