Rauzy tilings and bounded remainder sets on the torus

V. G. Zhuravlev

Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 83-106

Résumé

For the two dimensional torus $\mathbb{T}^2$ we construct the Rauzy tilings $d^0\supset d^1\supset\ldots\supset d^m\supset\ldots$, where each tiling $d^{m+1}$ turns out by inflation of $d^{m}$. The following results are proved: 1) Any tiling $d^{m}$ is invariant with respect to the shift $S(x)=x+\begin{pmatrix} \zeta \\ \zeta ^2\end{pmatrix}\mod\mathbb{Z}^2$, here $\zeta^{-1}> 1$ is a Pisot number satisfying the equation $x^3-x^2-x-1=0$. 2) The induced map $S^{(m)}=S|_{B^m d}$ is an exchange transformation of $B^m d\subset\mathbb{T}^2$, where $d=d^0$ and $B=\begin{pmatrix} - \zeta - \zeta \\ 1-\zeta ^2 \zeta^2\end{pmatrix}$. 3) The map $S^{(m)}$ is a shift of the torus $B^m d\simeq\mathbb{T}^2$ and $S^{(m)}$ is isomorphic to the initial shift $S$. It means that $d^m$ are infinite differentiable tilings. Let $Z_N(X)$ be equal to the number of points in the orbit $S^1(0), S^2(0)$, $\ldots,S^N(0)$ visited the domain $B^m d$. Then the remainder $r_N(B^md)=Z_N(B^m d)-N\zeta^m$ satisfies $-1.7$ for all $m$.