Rauzy tilings and bounded remainder sets on the torus

Abstract

For the two-dimensional torus $\mathbb{T}^2$, we construct the Rauzy tilings d⁰ ⊃ d¹ ⊃ … ⊃ d^m ⊃ …, where each tiling d^m+1 is obtained by subdividing the tiles of d^m. The following results are proved. Any tiling d^m is invariant with respect to the torus shift S(x) = x+ $\left( {_{\zeta ^2 }^\zeta } \right)$ mod ℤ², where ζ⁻¹ > 1 is the Pisot number satisfying the equation x³− x²−x-¹ = 0. The induced map $S^{(m)} = \left. S \right|_{B^m d}$ is an exchange transformation of B^md ⊂ $\mathbb{T}^2$, where d = d⁰ and $B = \left( {_{1 - \zeta ^2 \zeta ^2 }^{ - \zeta - \zeta } } \right)$ . The map S^(m) is a shift of the torus $B^m d \simeq \mathbb{T}^2$, which is affinely isomorphic to the original shift S. This means that the tilings d^m are infinitely differentiable. If Z_N(X) denotes the number of points in the orbit S¹(0), S²(0), …, S^N(0) belonging to the domain B^md, then, for all m, the remainder r_N(B^md) = Z_N(B^md) − N ζ^m satisfies the bounds −1.7 < r_N(B^md) < 0.5. Bibliography: 10 titles.