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On sums of powers of inverse complete quotients


Author: Oliver Jenkinson
Journal: Proc. Amer. Math. Soc. 136 (2008), 1023-1027
MSC (2000): Primary 26Dxx; Secondary 11A55, 37D20, 37E05
Published electronically: November 30, 2007
MathSciNet review: 2361877
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Abstract: For an irrational number $ x$, let $ x_n$ denote its $ n$-th continued fraction inverse complete quotient, obtained by deleting the first $ n$ partial quotients. For any positive real number $ r$, we establish the optimal linear bound on the sum of the $ r$-th powers of the first $ n$ complete quotients. That is, we find the smallest constants $ \alpha(r), \beta(r)$ such that $ x_1^r+\ldots +x_n^r< \alpha(r)n+\beta(r)$ for all $ n\ge1$ and all irrationals $ x$.


References [Enhancements On Off] (What's this?)

  • 1. I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433
  • 2. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909

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Additional Information

Oliver Jenkinson
Affiliation: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom
Email: omj@maths.qmul.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-07-09107-1
Received by editor(s): January 3, 2007
Published electronically: November 30, 2007
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.