Almost everywhere convergence of inverse fourier transforms

We show that if log(2 - Delta)f epsilon L-2(R-d), then the inverse Fourier transform of f converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when root(log(2 - Delta))f epsilon L-2(R-d) and log log(4 - Delta)f epsilon L-2(R-d). We also consider sequential convergence for general elements of L-2(R-d).