Tohoku Mathematical Journal
2018
June
SECOND SERIES VOL. 70, NO. 2

Tohoku Math. J.
70 (2018), 285-317

Title THREE CONSECUTIVE APPROXIMATION COEFFICIENTS: ASYMPTOTIC FREQUENCIES IN SEMI-REGULAR CASES

Author Jaap de Jonge and Cor Kraaikamp

(Received September 28, 2015, revised March 22, 2016)
Abstract. Denote by $p_n/q_n, n=1,2,3,\ldots,$ the sequence of continued fraction convergents of a real irrational number $x$. Define the sequence of approximation coefficients by $\theta_n(x):=q_n\left|q_nx-p_n\right|, n=1,2,3,\ldots$. In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as $\theta_{n-1}<\theta_n<\theta_{n+1}$, occur for almost all $x$ with only two different asymptotic frequencies. In this paper it is shown how these asymptotic frequencies can be determined for two other semi-regular cases. It appears that the optimal continued fraction has a similar distribution of only two asymptotic frequencies, albeit with different values. The six different values that are found in the case of the nearest integer continued fraction will show to be closely related to those of the optimal continued fraction.

Mathematics Subject Classification. Primary 11J70; Secondary 11K50.
Key words and phrases. Continued fractions, metric theory.

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