Structural and geometric characteristics of sets of convergence and divergence of multiple Fourier series with J k -lacunary sequence of rectangular partial sums

We study multiple trigonometric Fourier series of functions f in the classes , p > 1, which equal zero on some set (A mu is the Lebesgue measure), , N a parts per thousand yen 3. We consider the case when rectangular partial sums of the indicated Fourier series S (n) (x; f) have index n = (n (1), ..., n (N) ) a a"currency sign (N) , in which k (k a parts per thousand yen 1) components on the places {j (1), ..., j (k) } = J (k) aS, {1, ..., N} are elements of (single) lacunary sequences (i.e., we consider multiple Fourier series with J (k) -lacunary sequence of partial sums). A correlation is found of the number k and location (the "sample" J (k) ) of lacunary sequences in the index n with the structural and geometric characteristics of the set , which determines possibility of convergence almost everywhere of the considered series on some subset of positive measure of the set .2l