Abstract
Setokuchi and Takashima [7] gave general mathematical explanations for the emergence of several parabola-like hills in the behavior of the discrepancies of irrational rotations having single isolated large partial quotient, in somewhat short range of N. We extend estimates in [7] and give some general conditions which ensure repetitions of hills in much longer range of N. For example, in case of 1 − log10 7, our conditions show that more than 2.7 × 1027 repetitions of hills exist, where N runs up to over 6.8 × 1033.
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Setokuchi, T. On the discrepancy of irrational rotations with isolated large partial quotients: long term effects. Acta Math. Hungar. 147, 368–385 (2015). https://doi.org/10.1007/s10474-015-0563-0
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DOI: https://doi.org/10.1007/s10474-015-0563-0