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Mathematics of Computation

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On the speed of convergence of the nearest integer continued fraction

Author: H. Jager
Journal: Math. Comp. 39 (1982), 555-558
MSC: Primary 10K10
MathSciNet review: 669647
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Abstract: Let $ {p_n}/{q_n}$ and $ {A_n}/{B_n}$ denote the convergents of, respectively, the regular and the nearest integer continued fraction expansion of the irrational number x. There exists a function $ k(n)$ such that $ {A_n}/{B_n} = {p_{k(n)}}/{q_{k(n)}}$. Adams proved that for almost all x one has $ \lim k(n)/n = \log 2/\log G$, $ G = \frac{1}{2}(1 + \sqrt 5 )$. Here we present a shorter proof of this result, based on a simple expression for $ k(n)$ and the ergodicity of the shift operator, connected with the nearest integer continued fraction.

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Keywords: Nearest integer continued fraction, individual ergodic theorem
Article copyright: © Copyright 1982 American Mathematical Society

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