  We generalize the classical theorem by Jarnik and Besicovitch on the
irrationality exponents of real numbers and Hausdorff dimension. Let a be any
real number greater than or equal to 2 and let b be any non-negative real less
than or equal to 2/a. We show that there is a Cantor-like set with Hausdorff
dimension equal to b such that, with respect to its uniform measure, almost all
real numbers have irrationality exponent equal to a. We give an analogous
result relating the irrationality exponent and the effective Hausdorff
dimension of individual real numbers. We prove that there is a Cantor-like set
such that, with respect to its uniform measure, almost all elements in the set
have effective Hausdorff dimension equal to b and irrationality exponent equal
to a. In each case, we obtain the desired set as a distinguished path in a tree
of Cantor sets.
