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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


On a relationship between the convergents of the nearest integer and regular continued fractions

Author: William W. Adams
Journal: Math. Comp. 33 (1979), 1321-1331
MSC: Primary 10K10; Secondary 10K15
MathSciNet review: 537978
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Abstract: In this paper we derive a relation concerning the speed of convergence of the nearest integer and regular continued fractions. If $ {A_n}/{B_n}$, $ {p_k}/{q_k}$ denote the convergents of the nearest integer and regular continued fractions of an irrational number $ \alpha $, then for all n there is a $ k(n)$ such that $ {A_n}/{B_n} = {p_{k(n)}}/{q_{k(n)}}$. It is shown that

$\displaystyle \mathop {\lim }\limits_{n \to \infty } \;\frac{n}{{k\left( n \right)}} = \frac{{\log \left( {\frac{{1 + \sqrt 5 }}{2}} \right)}}{{\log \;2}}$

for almost all $ \alpha $. This problem is reduced to a special case of a general result concerning the frequency of partial quotients in the regular continued fraction (Theorem 2).

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PII: S 0025-5718(1979)0537978-9
Article copyright: © Copyright 1979 American Mathematical Society

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