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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.
11K06 General theory of distribution modulo \(1\)
11K31 Special sequences
11J70 Continued fractions and generalizations
Full Text: DOI
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