On sets of convergence and divergence of multiple orthogonal series

Multiple Fourier series with respect to uniformly bounded orthonormal systems (ONSs) are studied. The following results are obtained.
\medskip Theorem 1. \textit{Let $\Phi=\{\varphi_n(x)\}_{n=1}^\infty$ be a complete orthonormal system on $[0,1]$ that is uniformly bounded by $M$ on this interval, assume that $m\geqslant2$, and let $\Phi(m)=\{\varphi_{\mathbf n}(\mathbf x)\}_{\mathbf n \in\mathbb N^m}$, where $\varphi_{\mathbf n}(\mathbf n) =\varphi_{n_1}(x_1)\dotsb\varphi_{n_m}(x_m)$. Then there exists a function $f(\mathbf x)\in L([0,1]^m)$ cubically diverges on some measurable subset $\mathscr H$ of $[0,1]^m$ with $\mu_m(\mathscr H)\geqslant 1-(1-1/M^2)^m$. }
\medskip Theorem 3. For $M>1$ and an integer $m\geqslant 2$ let $E$ be an arbitrary measurable subset of $[0,1]$ such that $\mu(E)=1-1/M^2$. Then there exists a complete orthonormal system $\Phi$ on $[0,1]$ uniformly bounded by $M$ there such that the multiple Fourier series of each function $f(\mathbf x)\in L([0,1]^m)$ with respect to the product system $\Phi(m)$ cubically converges to $f(\mathbf x)$ a.e. on $E^m$.
\medskip Definitive results in this direction are established also for incomplete uniformly bounded ONSs.