Article ID Journal Published Year Pages File Type
4672994 Indagationes Mathematicae 2014 9 Pages PDF
Abstract

Denote by pn/qn,n=1,2,3,…pn/qn,n=1,2,3,…, the sequence of continued fraction convergents of the real irrational number xx. Define the sequence of approximation coefficients by θn:=qn|qnx−pn|,n=1,2,3,…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,…|θn+1−θn−1|,n=2,3,…, is simplified. The method involved also serves for showing that for almost all xx the pattern θn−1<θn<θn+1θn−1<θn<θn+1 occurs with the same asymptotic frequency as the pattern θn+1<θn<θn−1θn+1<θn<θn−1, namely 0.12109⋯0.12109⋯. All the four other patterns have the same asymptotic frequency 0.18945⋯0.18945⋯. The constants are explicitly given.

Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
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