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Rauzy tilings and bounded remainder sets on the torus. (English, Russian) Zbl 1158.11331
J. Math. Sci., New York 137, No. 2, 4658-4672 (2006); translation from Zap. Nauchn. Semin. POMI 322, 83-106 (2005).
Summary: For the two-dimensional torus $$\mathbb{T}^2$$, we construct the Rauzy tilings $$d_0\supset d_1 \supset\cdots\supset d_m \supset\dots$$, 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 $$S(x) = x+\binom{\zeta}{\zeta^2 } \bmod \mathbb Z_2$$, where $$\zeta^{-1}> 1$$ is the Pisot number satisfying the equation $$x^3-x^2-x-1 = 0.$$
The induced map $$S^{(m)} = S|_{B^md}$$ is an exchange transformation of $$B^md\subset\mathbb{T}^2$$, where $$d = d_0$$ and $$B = \left(\begin{smallmatrix} -\zeta & -\zeta\\ 1-\zeta^2 & \zeta^2 \end{smallmatrix}\right)$$.
The map $$S(m)$$ is a shift of the torus $$B^md\simeq \mathbb{T}^2$$, 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 $$S^1(0),$$ $$S^2(0),\dots, S^N(0)$$ belonging to the domain $$B^md$$, then, for all $$m$$, the remainder $$r_N(B^md) = Z_N(B^md)-N\zeta^m$$ satisfies the bounds $$-1.7 < r_N(B^md) < 0.5$$.

MSC:
 11K99 Probabilistic theory: distribution modulo $$1$$; metric theory of algorithms 11J71 Distribution modulo one 37B99 Topological dynamics
Keywords:
Rauzy tilings; cubic Pisot number
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References:
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