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Moduli of toric tilings into bounded remainder sets and balanced words


Author: V. G. Zhuravlev
Translated by: N. V. Tsilevich
Original publication: Algebra i Analiz, tom 24 (2012), nomer 4.
Journal: St. Petersburg Math. J. 24 (2013), 601-629
MSC (2010): Primary 52C22; Secondary 37B50
DOI: https://doi.org/10.1090/S1061-0022-2013-01256-8
Published electronically: May 24, 2013
MathSciNet review: 3088009
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Abstract: The moduli space $ \mathcal {M}_{\mathrm {til}}$ is constructed for the family $ \mathbb{T}_{\mathrm {til}}$ of parallelotope tilings

$\displaystyle \mathbb{T}^{D}_{c,\lambda }=\mathbb{T}^{D}_0 \sqcup \mathbb{T}^{D}_1 \sqcup \dots \sqcup \mathbb{T}^{D}_D $

of the torus $ \mathbb{T}^D=\mathbb{R}^D/\mathbb{Z}^D$ of arbitrary dimension $ D$ into bounded remainder sets $ \mathbb{T}^{D}_k$. By using these tilings, the Hecke theorem on the distribution of fractional parts on the circle is extended to the tori $ \mathbb{T}^D$: the deviation of the distribution of points of an orbit with respect to the translation $ S_{\beta }\,:\, x \rightarrow x+\beta \bmod \mathbb{Z}^D$ of the torus $ \mathbb{T}^D$ by an arbitrary vector $ \beta =\frac {1}{n}(\lambda c+l)$ is estimated in terms of the moduli $ (c,\lambda )\in \mathcal {M}_{\mathrm {til}}$, where $ l$ lies in the cubic lattice $ \mathbb{Z}^D$.

The color and frequency universality is proved for the toric tilings $ \mathbb{T}^{D}_{c,\lambda }$ from the family $ \mathbb{T}_{\mathrm {til}}$ and it is shown how these tilings can be used to generate $ \kappa $-balanced words $ w$ in the alphabet $ \mathcal {A}=\{0,1, \dots ,D \}$ with $ \kappa =2$ for $ D=2$ and $ \kappa =3$ for $ D\geq 3$.


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Additional Information

V. G. Zhuravlev
Affiliation: Vladimir State Humanitarian University, pr. Stroiteleǐ 11, Vladimir 600024, Russia
Email: vzhuravlev@mail.ru

DOI: https://doi.org/10.1090/S1061-0022-2013-01256-8
Keywords: Hecke theorem, distribution of fractional parts, bounder remainder sets on the torus
Received by editor(s): December 20, 2010
Published electronically: May 24, 2013
Additional Notes: Supported by RFBR (grant no. 11-01-00578-a)
Article copyright: © Copyright 2013 American Mathematical Society