Metric discrepancy results for geometric progressions perturbed by irrational rotations

  • K. FukuyamaEmail author
  • S. Mori
  • Y. Tanabe


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.

Key words and phrases

discrepancy lacunary sequence law of the iterated logarithm 

Mathematics Subject Classification

primary 11K38 42A55 60F15 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsKobe UniversityRokko, KobeJapan

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