Bounded remainder sets on the double covering of the Klein bottle

V. G. Zhuravlev

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 29, Tome 429 (2014), pp. 82-105

Résumé

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.