Almost everywhere convergence of inverse fourier transforms

We show that if log(2 - Δ)f ∞ L²(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 √log(2 - Δ)f ∈ L₂(ℝ^d) and log log(4 - Δ)f ∈ L₂(ℝ^d). We also consider sequential convergence for general elements of L²(ℝ ^d).