On universality and convergence of the Fourier series of functions in the disc algebra

We construct functions in the disc algebra with pointwise universal Fourier series on sets which are G-delta and dense and at the same time with Fourier series whose set of divergence is of Hausdorff dimension zero. We also see that some classes of closed sets of measure zero do not accept uniformly universal Fourier series, although all such sets accept divergent Fourier series.