Coherence on Fractals Versus Pointwise Convergence for the Schrodinger Equation

We consider Carleson's problem regarding convergence for the Schrodinger equation in dimensions d >= 2. We show that if the solution converges almost everywhere with respect to alpha-Hausdorff measure to its initial datum as time tends to zero, for all data H-s (R-d), then s >= d/2(d+2)( d + 1 - alpha). This strengthens and generalises results of Bourgain and Dahlberg-Kenig.