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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the speed of convergence of the nearest integer continued fraction
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by H. Jager PDF
Math. Comp. 39 (1982), 555-558 Request permission

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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Math. Comp. 39 (1982), 555-558
  • MSC: Primary 10K10
  • DOI: https://doi.org/10.1090/S0025-5718-1982-0669647-9
  • MathSciNet review: 669647