The structure and geometry of maximal sets of convergence and unbounded divergence almost everywhere of multiple Fourier series of functions in $L_1$ equal to zero on a given set

The precise structure and geometry of maximal sets of convergence and unbounded divergence almost everywhere (a.e.) of Fourier series of functions in the class $L_1(T^N)$, $N\geqslant1$, $T^N[0,2\pi]^N$, and vanishing on a given measurable set $E$ is found (in the case $N\geqslant2$ this is done for both rectangular and square summation).
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