On the approximation by three consecutive continued fraction convergents |
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Authors: | Hendrik Jager Jaap de Jonge |
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Affiliation: | 1. Oude Larenseweg 26, 7214 PC Epse, The Netherlands;2. Von Zesenstraat 172, 1093 BJ Amsterdam, The Netherlands |
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Abstract: | Denote by pn/qn,n=1,2,3,…, the sequence of continued fraction convergents of the real irrational number x. Define the sequence of approximation coefficients by θn:=qn|qnx−pn|,n=1,2,3,…. A laborious way of determining the mean value of the sequence |θn+1−θn−1|,n=2,3,…, is simplified. The method involved also serves for showing that for almost all x the pattern θn−1<θn<θn+1 occurs with the same asymptotic frequency as the pattern θn+1<θn<θn−1, namely 0.12109?. All the four other patterns have the same asymptotic frequency 0.18945?. The constants are explicitly given. |
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Keywords: | Continued fractions Metric theory |
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