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On the one-sided boundedness of sums of fractional parts \((\{ n\alpha+ \gamma\}-\frac12)\). (English) Zbl 1049.11086

Summary: Given an irrational \(\alpha\) in \([0, 1)\), we ask for which values of \(\gamma\) in \([0, 1)\) the sums \[ C(m,\alpha,\gamma):= \sum_{n=1}^ m(\{n\alpha+\gamma\}-\tfrac12) \] are bounded from above or from below for all \(m\). When the partial quotients in the continued fraction expansion of \(\alpha=[0, a_1, a_2, \dots]\) are bounded, say \(a_i\leq A\), we give a necessary condition on \(\gamma\) (involving the non-homogeneous continued fraction expansion of \(\gamma\) with respect to \(\alpha\)). When the \(a_i\geq 2\) we give examples of \(\gamma\) that cause one-sided boundedness. In particular, when \(2\leq a_i\leq A\) and the \(a_{2i-1}\) (respectively \(a_{2i}\)) are all even, we can deduce that \(C(m, \alpha, \gamma)\) is bounded from below (resp. above) if and only if \(\gamma=\{\frac12 \alpha, s\alpha\}\) (resp. \(\gamma=\{\frac12 \alpha+ s\alpha\}\)) for some integer \(s\). The sums \(|C(m, \alpha, \gamma)|\) are always unbounded with \(| C(m, \alpha, \gamma)| >c \log m\) for infinitely many \(m\).
This paper continues the author’s recent article [ibid. 65, 48–73 (1997; Zbl 0886.11045)].

MSC:

11K31 Special sequences
11J70 Continued fractions and generalizations
11K38 Irregularities of distribution, discrepancy

Citations:

Zbl 0886.11045
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References:

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