Metric discrepancy results for geometric progressions perturbed by irrational rotations


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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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).

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Key words and phrases

  • discrepancy
  • lacunary sequence
  • law of the iterated logarithm

Mathematics Subject Classification

  • primary 11K38
  • 42A55
  • 60F15