A weak generalized localization criterion for multiple Walsh–Fourier series with $J_k$-lacunary sequence of rectangular partial sums

We obtain a criterion for the validity of weak generalized localization almost everywhere on an arbitrary set of positive measure $\mathfrak A$, $\mathfrak A\subset\mathbb I^N=\{x\in\mathbb R^N\colon0\leq x_j<1,\, j=1,2,\dots,N\}$, $N\geq3$ (in terms of the structure and geometry of the set $\mathfrak A$), for multiple Walsh–Fourier series (summed over rectangles) of functions $f$ in the classes $L_p(\mathbb I^N)$, $p>1$ (i.e., necessary and sufficient conditions for the convergence almost everywhere of the Fourier series on some subset of positive measure $\mathfrak A_1$ of the set $\mathfrak A$, when the function expanded in a series equals zero on $\mathfrak A$), in the case when the rectangular partial sums $S_n(x;f)$ of this series have indices $n=(n_1,\dots,n_N)\in\mathbb Z^N$ in which some components are elements of (single) lacunary sequences.