Sufficient conditions for the convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in the classes $L_p$, $p>1$, are obtained in the case where rectangular partial sums $S_n(x;f)$ of this series have numbers $n=(n_1,\dots,n_N)\in \mathbb Z^N$, $N\geq 3$, such that of $N$ components only $k$ ($1\leq k\leq N-2$) are elements of some lacunary sequences. Earlier, in the case where $N-1$ components of the number $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained for functions in the classes $L_p$, $p>1$, by M. Kojima (1977), and for functions in Orlizc classes by D. K. Sanadze, Sh. V. Kheladze (1977) and N. Yu. Antonov (2014). Note that presence of two or more “free” components in the number $n$, as follows from the results by Ch. Fefferman (1971) and M. Kojima (1977), does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $N\geq 3$ even in the class of continuous functions.
"Sufficient Conditions for Convergence Almost Everywhere of Multiple Trigonometric Fourier Series with Lacunary Sequence of Partial Sums." Real Anal. Exchange 41 (1) 159 - 172, 2016.