Bounded remainder sets on the double covering of the Klein bottle

The shift $\widetilde{\mathbb S}\colon\widetilde{\mathbb K}^2\to\widetilde{\mathbb K}^2$ on the double covering of the Klein bottle $\widetilde{\mathbb K}^2=\mathbb K^2\times\{\pm1\}$ is considered. This shift $\widetilde{\mathbb S}$ generates some tiling $\widetilde{\mathbb K}^2=\widetilde{\mathbb K}^2_0\sqcup\widetilde{\mathbb K}^2_1$ into two bounded remainder sets $\widetilde{\mathbb K}^2_0$ and $\widetilde{\mathbb K}^2_1$ with respect to the shift $\widetilde{\mathbb S}$. Two-sided estimates are proved for the deviation functions of these sets.