Abstract
For an irrational \(\alpha \), we investigate the sums \(\sum _{i=1}^n \left( \{i \alpha \} - \frac{1}{2} \right) \) and \(\sum _{i=1}^n \left\{ \left( \{i \alpha \} - \frac{1}{2} \right) ^2 - \frac{1}{12} \right\} \). We discuss exact formulae and asymptotic estimates for these sums and point out interesting geometrical properties of their graphs in the case when the continued fraction expansion of \(\alpha \) has a large isolated partial quotient.
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Mori, Y., Shimaru, N. & Takashima, K. On the distribution of partial sums of irrational rotations. Period Math Hung 78, 88–97 (2019). https://doi.org/10.1007/s10998-018-00273-y
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DOI: https://doi.org/10.1007/s10998-018-00273-y