Localization for Multiple Fourier Series with "$J_k$-Lacunary Sequence of Partial Sums" in Orlicz Classes

We obtain structural and geometric characteristics of sets on which weak generalized localization almost everywhere is valid for multiple trigonometric Fourier series of functions in the classes $L(\log^+L)^{3k+2}(\mathbb T^N)$, $1\le k\le N-2$, $N\ge 3$, in the case where the rectangular partial sums of these series have a “number” in which exactly $k$ components are terms of lacunary sequences.