Hausdorff dimension in inhomogeneous diophantine approximation

Let α be an irrational real number. We show that the set of ϵ-badly approximable numbers has full Hausdorff dimension for some positive ϵ if and only if α is singular on average. The condition is equivalent to the average 1/k Σ i=1,••• ,k log ai of the logarithms of the partial quotients ai of α going to infinity with k. We also consider one-sided approximation, obtain a stronger result when ai tends to infinity, and establish a partial result in higher dimensions.