Analogs of the Luzin–Danzhua and Cantor–Lebesgue theorems for double trigonometric series

Let $\|\cdot\|$ be some norm in $R^2$, $\Gamma$ be the unit sphere induced in $R^2$ by this norm, and $\{A_j\}$ a sequence of disjoint subsets of $R_+$ such that if $\nu\in A_j$, then $\nu\cdot\Gamma\cap Z^N\ne\varnothing$. For series of the form
$$\sum_{j=1}^\infty\sum_{\|n\|\in A_j}c_ne^{2\pi i(n_1x_1+n_2x_2)}$$
analogs of the Luzin–Danzhu and Cantor–Lebesgue theorems are established.