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On sums of powers of inverse complete quotients. (English) Zbl 1130.26011

Denoting the expansion of an irrational number \(x\) into an infinite fraction by \(x=[a_0; a_1, a_2, \dots[,\) its \(k\)-th inverse complete quotient is \(x_k=[0; a_k, a_{k+1},\dots[.\) The author offers proof that \(\sum^n_{k=1}x^r_k\) is \(< (n-1)c^r+1\) if \(0 <r \leq -\log_c 2\) and is \(<(n+1)/2\) if \(-\log_c 2 < r ,\) where \(c=(5^{1/2}-1)/2;\) and that the bounds are sharp in a sense.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
11A55 Continued fractions
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37E05 Dynamical systems involving maps of the interval
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References:

[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. · Zbl 0493.28007
[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. · Zbl 0423.10001
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