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On discrepancies of irrational rotations: an approach via rational rotation

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Abstract

Setokuchi and Takashima (Unif Distrib Theory (2) 9:31–57, 2014) and Setokuchi (Acta Math Hung, [11]) gave refinements of estimates for discrepancies by using Schoissengeier’s exact formula. Mori and Takashima (Period Math Hung, [7]) discussed the distribution of the leading digits of \(a^n\) by approximating irrational rotations by “rational rotations”. We apply here their methods to the estimation of discrepancies. We give much more accurate estimates for discrepancies by simple direct calculations, without using Schoissengeier’s formula. We show that the initial segment of the graph of discrepancies of irrational rotations with a single isolated large partial quotient is linearly decreasing, provided we observe the discrepancies on a linear scale with suitable step. We also prove that large hills, caused by single isolated large partial quotients, will appear infinitely often.

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Acknowledgements

The author would like to express his hearty thanks to the referee and the editor for their helpful advices and fruitful discussions on our problems.

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Authors and Affiliations

  1. Department of Applied Mathematics, Graduate School of Science, Okayama University of Science, 1-1 Ridai-cho, Okayama, 700-0005, Japan

    Naoto Shimaru

  2. Department of Applied Mathematics, Okayama University of Science, 1-1 Ridai-cho, Okayama, 700-0005, Japan

    Keizo Takashima

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  1. Naoto Shimaru
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  2. Keizo Takashima
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Correspondence to Keizo Takashima.

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Shimaru, N., Takashima, K. On discrepancies of irrational rotations: an approach via rational rotation. Period Math Hung 75, 29–35 (2017). https://doi.org/10.1007/s10998-016-0164-x

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  • DOI: https://doi.org/10.1007/s10998-016-0164-x

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