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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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