Multidimensional Hecke theorem on the distribution of fractional parts
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V. G. Zhuravlev
Translated by: A. Luzgarev - St. Petersburg Math. J. 24 (2013), 71-97
- DOI: https://doi.org/10.1090/S1061-0022-2012-01232-X
- Published electronically: November 15, 2012
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Abstract:
Hecke’s theorem on the distribution of fractional parts on the unit circle is generalized to the tori $\mathbb {T}^D=\mathbb {R}^D/L$ of arbitrary dimension $D$. It is proved that $|\delta _{k}(i)| \leq c_k n$ for $i=0,1,2,\dots$, where $\delta _{k}(i)=r_{k}(i) -ia_k$ is the deviation of the number $r_{k}(i)$ of returns in $i$ steps into $\mathbb {T}_k^D \subset \mathbb {T}^D$ for the points of an $S_{\beta }$-orbit from its mean value $a_k= \mathrm {vol}(\mathbb {T}_k^D)/\mathrm {vol}(\mathbb {T}^D)$, where $\mathrm {vol}(\mathbb {T}_k^D)$ and $\mathrm {vol}(\mathbb {T}^D)$ denote the volumes of the tile $\mathbb {T}_k^D$ and of the torus $\mathbb {T}^D$. The tiles $\mathbb {T}_k^D$ in question have the following property: for the torus $\mathbb {T}^D$ there exists a development $T^D \subset \mathbb {R}^D$ such that a shift $S_{\alpha }$ of the torus $\mathbb {T}^D$ is equivalent to some exchange transformation of the corresponding tiles $T_k^D$ in a partition of the development $T^D= T_0^D \sqcup T_1^D \sqcup \dots \sqcup T_D^D$. The torus shift vectors $S_{\alpha }$, $S_{\beta }$ satisfy the condition $\alpha \equiv n \beta \bmod L$, where $n$ is any natural number, and the constants $c_k$ in the inequalities are expressed in terms of the diameter of the development $T^D$.References
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Bibliographic Information
- V. G. Zhuravlev
- Affiliation: Vladimir State University for the Humanities, pr. Stroiteley 11, Vladimir 600024, Russia
- Email: vzhuravlev@mail.ru
- Received by editor(s): December 20, 2010
- Published electronically: November 15, 2012
- Additional Notes: Supported by RFBR (grant no. 11-01-00578-a)
- © Copyright 2012 American Mathematical Society
- Journal: St. Petersburg Math. J. 24 (2013), 71-97
- MSC (2010): Primary 11K60, secondary, 11H06
- DOI: https://doi.org/10.1090/S1061-0022-2012-01232-X
- MathSciNet review: 3013295