Given a compact metric space (Q,d) equipped with a non-atomic, probability
measure m and a positive decreasing function ip, we consider a natural class of limsup
subsets A(ip) of 0. The classical limsup set W(ip) of '^-approximable' numbers in
the theory of metric Diophantine approximation fall within this class. We establish
sufficient conditions (which are also necessary under some natural assumptions) for
the m-measure of A(ip) to be either positive or full in Q and for the Hausdorff f-
measure to be infinite. The classical theorems of Khintchine-Groshev and Jarnik
concerning W{iji) fall into our general framework. The main results provide a unifying
treatment of numerous problems in metric Diophantine approximation including those
for real, complex and p-adic fields associated with both independent and dependent
quantities. Applications also include those to Kleinian groups and rational maps.
Compared to previous works our framework allows us to successfully remove many
unnecessary conditions and strengthen fundamental results such as Jarnik's theorem
and the Baker-Schmidt theorem. In particular, the strengthening of Jarnik's theorem
opens up the Duffm-Schaeffer conjecture for Hausdorff measures.
Mathematics Subject Classification: 11J83; 11313, 11K60, 28A78, 28A80
Keywords: Metric Diophantine approximation, Hausdorff measure and dimension,
limsup sets, Khintchine and 3arnik theorems, zero-one laws,
Received by the editor September 14 2004
V Beresnevich's work was partially supported by INTAS project 00-429
S Velani is a Royal Society University Research Fellow
Previous Page Next Page