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Embeddings of circular orbits and the distribution of fractional parts. (English. Russian original) Zbl 1354.11052
St. Petersbg. Math. J. 26, No. 6, 881-909 (2015); translation from Algebra Anal. 26, No. 6, 29-68 (2014).
Summary: 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 $$|\delta_{n,\alpha}(i,t)|\leq | n|$$ 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 $$|\delta_{n,\alpha}(i, t)|< c_{\alpha}$$ for a constant $$c_{\alpha}>0$$ depending on $$\alpha$$, as $$| n| \rightarrow \infty$$ 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.
##### MSC:
 11K06 General theory of distribution modulo $$1$$ 11K31 Special sequences 11J70 Continued fractions and generalizations
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##### References:
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