On almost everywhere convergence for lacunary sequences of multiple rectangular Fourier sums

Let a sequence of $d$-dimensional vectors $\mathbf{n}_k=(n_k^1, n_k^2,\ldots,n_k^d)$ with positive integer coordinates satisfy the condition $n_k^j=\alpha_j m_k+O(1), \ k \in {\mathbb N}, \ 1 \le j \le d,$\; where $\alpha _1>0,$ $\ldots,\alpha _d>0,$ and $\{ m_k \} _{k=1}^{\infty }$ is an increasing sequence of positive integers. Under some conditions on a function $\varphi :[0,+\infty ) \to [0,+\infty )$, it is proved that, if the sequence of Fourier sums $S_{m_k}(g,x)$ converges almost everywhere for any function $g \in \varphi (L) ([0 , 2\pi ))$, then, for any $d \in {\mathbb N}$ and $f \in \varphi (L)(\ln ^+L)^{d-1}([0 , 2\pi ) ^d) $, the sequence $ S_{\mathbf {n}_k} (f,\mathbf x)$ of rectangular partial sums of the multiple trigonometric Fourier series of the function $f$ and the corresponding sequences of partial sums of all conjugate series converge almost everywhere.