Necessary Conditions for the Weak Generalized Localization of Fourier Series with “Lacunary Sequence of Partial Sums”

It has been established that, on the subsets $\mathbb{T}^N=[-\pi,\pi]^N$ describing a cross $W$ composed of $N$-dimensional blocks, $W_{x_sx_t}=\Omega_{x_sx_t}\times [-\pi,\pi]^{N-2}$ ($\Omega_{x_sx_t}$ is an open subset of $[-\pi,\pi]^2$) in the classes $L_p(\mathbb{T}^N)$, $p>1$, a weak generalized localization holds, for $N\ge3$, almost everywhere for multiple trigonometric Fourier series when to the rectangular partial sums $S_n(x;f)$ ($x\in\mathbb{T}^N$, $f\in L_p$) of these series corresponds the number $n=(n_1,\dots,n_N)\in\mathbb Z_{+}^{N}$ some components $n_j$ of which are elements of lacunary sequences. In the present paper, we prove a number of statements showing that the structural and geometric characteristics of such subsets are sharp in the sense of the numbers (generating $W$) of the $N$-dimensional blocks $W_{x_sx_t}$ as well as of the structure and geometry of $W_{x_sx_t}$. In particular, it is proved that it is impossible to take an arbitrary measurable two-dimensional set or an open three-dimensional set as the base of the block.