Jenkinson, Oliver On sums of powers of inverse complete quotients. (English) Zbl 1130.26011 Proc. Am. Math. Soc. 136, No. 3, 1023-1027 (2008). 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. Reviewer: János Aczél (Waterloo/Ontario) Cited in 2 Documents 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 Keywords:sequences; infinite fractions; inverse complete quotient; inequalities; sharp bounds; Lebesgue almost every irrational PDFBibTeX XMLCite \textit{O. Jenkinson}, Proc. Am. Math. Soc. 136, No. 3, 1023--1027 (2008; Zbl 1130.26011) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.