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A generalization of the Gauss–Kuzmin–Wirsing constant

Abstract

The Gauss–Kuzmin–Wirsing constant is generalized to the transformation \(T_p(x)=\left\{ \dfrac{p}{x}\right\} \), where p is any given positive integer and \(\left\{ \dfrac{p}{x}\right\} \) denotes the fractional part of \(\dfrac{p}{x}\). A near-optimal estimate for the generalized constant is obtained.

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Acknowledgements

The author is supported by National Natural Science Foundation of China (No. 11571387) and CUFE Young Elite Teacher Project (No. QYP1902).

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Correspondence to Peng Sun.

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Sun, P. A generalization of the Gauss–Kuzmin–Wirsing constant. Monatsh Math (2021). https://doi.org/10.1007/s00605-021-01622-9

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Keywords

  • Gauss transformation
  • Continued fraction
  • Transfer operator
  • Gauss’ problem
  • Wirsing’s constant

Mathematics Subject Classification

  • Primary: 11J70
  • 11K50
  • 37C30