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We consider the class of multiple Fourier series associated with functions inthe Dirichlet space of the polydisc. We prove that every such series issummable with respect to unrestricted rectangular partial sums, everywhereexcept for a set of zero multi-parametric logarithmic capacity. Conversely,given a compact set in the torus of zero capacity, we construct a Fourierseries in the class which diverges on this set, in the sense of Pringsheim. Wealso prove that the multi-parametric logarithmic capacity characterizes theexceptional sets for the radial variation and radial limits of Dirichlet spacefunctions. As a by-product of the methods of proof, the results also hold inthe vector-valued setting.

Rectangular summation of multiple Fourier series and multi-parametric capacity