On asymptotics of Dirichlet's kernels of Fourier sums with respect to a polygon

Given a polygon $W\in\mathbb R^2$, we study the behaviour of two-dimensional Dirichlet's kernels $D_{RW}(x,y)=\sum_{(n,m)\in RW}e^{-2\pi i(nx+my)}$ as $R\to+\infty$. It is well-known that $\|D_{RW}\|_{L([-1/2,1/2]^2)}\asymp\ln^2R$ for any polygon $W$ and that $\|D_{RW}-\hat\chi_{RW}\|=O(\ln R)$ if the coordinates of the vertices of $W$ are rational. We show that in general the second assertion does not hold. Namely, there is such a triangle $W$ that $\varlimsup_{R\to+\infty}\frac1{\ln^2R}(\|D_{RW}\|-\|\hat\chi_{RW}\|)>0$.