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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
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