Necessary conditions for the weak generalized localization of Fourier series with “lacunary sequence of partial sums”

It has been established that, on the subsets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{T}^N = [ - \pi ,\pi ]^N $$\end{document} describing a cross W composed of N-dimensional blocks, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ W_{x_s x_t } = \Omega _{x_s x_t } \times [ - \pi ,\pi ]^{N - 2} (\Omega _{x_s x_t } $$\end{document} is an open subset of [−π, π]2) in the classes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L_p (\mathbb{T}^N ),p > 1 $$\end{document}, a weak generalized localization holds, for N ≥ 3, almost everywhere for multiple trigonometric Fourier series when to the rectangular partial sums \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ S_n (x;f)(x \in \mathbb{T}^N ,f \in L_p ) $$\end{document} of these series corresponds the number n = (n1,…, nN) ∈ ℤ+N, some components nj 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ W_{x_s x_t } $$\end{document} as well as of the structure and geometry of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ W_{x_s x_t } $$\end{document}. 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.