On the Cantor–Lebesgue theorem for double trigonometric series

Suppose that on some measurable set $E\subset\mathbf{T}^2$, $\mu(E)>2/3$,
$$ A_\nu(x)=\sum_{n_1^2+n_2^2=\nu}c_{n_1,n_2}e^{2\pi i(n_1x_1+n_2x_2)}\to0\qquad(\nu\to\infty). $$
Then
$$ \sum_{n_1^2+n_2^2=\nu}|c_{n_1,n_2}|^2\to0\qquad(\nu\to\infty). $$