Hausdorff dimension and uniform exponents in dimension two

In this paper we prove that the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent mu in (1/2, 1) is equal to 2(1 - mu) for mu >= v 2/2, whereas for mu < v 2/2 it is greater than 2(1 - mu) and at most equal to (3 - 2 mu)(1 - mu)/(1 -mu + mu(2)). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when mu tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. Moreover, we prove a lower bound for the packing dimension, which appears to be strictly greater than the Hausdorff dimension for mu >= 0.565....