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Metric discrepancy results for geometric progressions perturbed by irrational rotations

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Abstract

For \(\theta \in (-\infty , -1)\cup (1, \infty )\) and for almost every x, it is known that the sequence \(\{\theta ^k x\}\) is uniformly distributed modulo 1. The speed of convergence sensitively depends on the algebraic nature of \({\theta}\). In this paper we prove that such dependence vanishes if we perturb the sequence by adding the irrational rotation \(\{\kappa\gamma\}\). The speed becomes identical with that of the sequence of uniformly distributed independent random variables.

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Authors and Affiliations

  1. Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501, Japan

    K. Fukuyama, S. Mori & Y. Tanabe

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  1. K. Fukuyama
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  2. S. Mori
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  3. Y. Tanabe
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Correspondence to K. Fukuyama.

Additional information

The first author is supported by JSPS KAKENHI 16K05204.

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Fukuyama, K., Mori, S. & Tanabe, Y. Metric discrepancy results for geometric progressions perturbed by irrational rotations. Acta Math. Hungar. 161, 48–65 (2020). https://doi.org/10.1007/s10474-019-01003-7

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  • DOI: https://doi.org/10.1007/s10474-019-01003-7

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