Cantor–Lebesgue theorem for double trignometric series

Let $||\cdot||$ be a norm in $\mathbf{R}^2$ and let $\Gamma$ be the unit sphere induced by this norm. We call a segment joining points $x, y\in\mathbf{R}^2$ rational if $(x_1-y_1)/(x_2-y_2)$ или $(x_2-y_2)/(x_1-y_1)$ is a rational number. Let $\Gamma$ be a convex curve containing no rational segments. Satisfaction of the condition
$$ T_\nu(x)=\sum_{||n||=\nu}c_n e^{2\pi i(n_1x_1+n_2x_2)}\to0\qquad (\nu\to\infty) $$
in measure on the set $E\subset[-\frac12, \frac12)\times[-\frac12, \frac12)=T^2$ of positive planar measure implies $||T_\nu||_{L_4}(T^2)\to0$ ($\nu\to\infty$). If, however, $\Gamma$ contains a rational segment, then there exist a sequence of polynomials $\{T_\nu\}$ and a set $E\subset T^2$, $|E|>0$, such that $T_\nu(x)\to0$ ($\nu\to\infty$) on $E$; however, $|c_n|\not\to0$ for $||n||\to\infty$.