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Embeddings of circular orbits and the distribution of fractional parts

Author: V. G. Zhuravlev
Translated by: N. V. Tsilevich
Original publication: Algebra i Analiz, tom 26 (2014), nomer 6.
Journal: St. Petersburg Math. J. 26 (2015), 881-909
MSC (2010): Primary 11K06
Published electronically: September 21, 2015
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Abstract: Let $ r_{n,\alpha } (i,t)$ be the number of points of the sequence $ \{t\}, \{\alpha +t\}, \{2\alpha +t\},\dots $ that fall into the semiopen interval $ [0, \{n\alpha \})$, where $ \{x\}$ is the fractional part of $ x$, $ n$ is an arbitrary integer, and $ t$ is any fixed number. Denote by $ \delta _{n,\alpha }(i,t)= i \{n \alpha \} - r_{n,\alpha } (i,t)$ the deviation of the expected number $ i \{n \alpha \}$ of hits of the above sequence in the semiopen interval $ [0, \{n\alpha \})$ of length $ \{n \alpha \}$ from the observed number of hits $ r_{n,\alpha } (i,t)$. E. Hecke proved the following theorem: the deviations $ \delta _{n,\alpha }(i,t)$ satisfy the inequality $ \vert\delta _{n,\alpha }(i,t)\vert\le \vert n\vert$ for all $ t\in [0,1)$ and $ i=0,1,2,\dots $. In this paper, conditions on the parameters $ n$ and $ \alpha $ are found under which $ \delta _{n,\alpha }(i, t)$ can be bounded as $ \vert\delta _{n,\alpha }(i, t)\vert< c_{\alpha }$ for a constant $ c_{\alpha }>0$ depending on $ \alpha $, as $ \vert n\vert \rightarrow \infty $ and $ n$ ranges over an infinite subset of integers. In the case where $ n$ is taken to be equal to the denominators of the convergents $ Q_m$ to $ \alpha $, the smallest values of the constants $ c_{\alpha }$ are computed. The proofs involve a new method based on embeddings of circular orbits into partitions of the unit circle.

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Additional Information

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

Keywords: Hecke theorem, distribution of fractional parts, bounded remainder sets
Received by editor(s): June 25, 2013
Published electronically: September 21, 2015
Additional Notes: Supported by RFBR (grant no. 4-01-00360)
Article copyright: © Copyright 2015 American Mathematical Society