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Multidimensional Hecke theorem on the distribution of fractional parts

Author: V. G. Zhuravlev
Translated by: A. Luzgarev
Original publication: Algebra i Analiz, tom 24 (2012), nomer 1.
Journal: St. Petersburg Math. J. 24 (2013), 71-97
MSC (2010): Primary 11K60, 11H06
Published electronically: November 15, 2012
MathSciNet review: 3013295
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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 $ \vert\delta _{k}(i)\vert \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$.

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Additional Information

V. G. Zhuravlev
Affiliation: Vladimir State University for the Humanities, pr. Stroiteley 11, Vladimir 600024, Russia

Keywords: Hecke theorem, fractional parts distribution, mean values of deviation functions, bounded remainder sets on the torus
Received by editor(s): December 20, 2010
Published electronically: November 15, 2012
Additional Notes: Supported by RFBR (grant no. 11-01-00578-a)
Article copyright: © Copyright 2012 American Mathematical Society

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