Growth rate of sequences of multiple rectangular Fourier sums

In the case when a sequence of $d$-dimensional vectors $\mathbf n_k=(n_k^1,n_k^2,\dots,n_k^d)$ with nonnegative integral coordinates satisfies the condition
$$ n_k^j=\alpha_jm_k+O(1),\quad k\in\mathbb N,\quad1\le j\le d, $$
where $\alpha_1\dots,\alpha_d$ are nonnegative real numbers and $\{m_k\}_{k=1}^\infty$ is a sequence of positive integers, the following estimate of the rate of growth of sequences $S_{\mathbf n_k}(f,\mathbf x)$ of rectangular partial sums of multiple trigonometric Fourier series is obtained: if $f\in L(\ln^+L)^{d-1}([-\pi,\pi)^d)$, then
$$ S_{\mathbf n_k}(f,\mathbf x)=o(\ln k)\quad\text{a.e.} $$
Analogous estimates are valid for conjugate series as well.