On the Divergence Sets of Fourier Series in Systems of Characters of Compact Abelian Groups

For a class of character systems of compact Abelian groups and for homogeneous Banach spaces $B$ satisfying some additional regularity conditions, we prove the following alternative: either the Fourier series of an arbitrary function in $B$ converges almost everywhere, or there exists a function in $B$ whose Fourier series diverges everywhere. We also prove that the classes of divergence sets of Fourier series in such function systems in the above-mentioned spaces are closed under at most countable unions and contain all sets of measure zero. As corollaries, we obtain some well-known and new results on everywhere divergent Fourier series in the trigonometric system as well as in the Walsh and Vilenkin systems and their rearrangements.