## Tohoku Mathematical Journal

### Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases

#### 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 70, Number 2 (2018), 285-317.

Dates
First available in Project Euclid: 2 June 2018

https://projecteuclid.org/euclid.tmj/1527904823

Digital Object Identifier
doi:10.2748/tmj/1527904823

Mathematical Reviews number (MathSciNet)
MR3810242

Zentralblatt MATH identifier
06929336

Keywords
Continued fractions metric theory

#### Citation

Jonge, Jaap de; Kraaikamp, Cor. Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases. Tohoku Math. J. (2) 70 (2018), no. 2, 285--317. doi:10.2748/tmj/1527904823. https://projecteuclid.org/euclid.tmj/1527904823

#### References

• W. Bosma, Optimal continued fractions, Nederl. Akad. Wetensch. Indag. Math. 49 (1987), no. 4, 353–379.
• W. Bosma and C. Kraaikamp, Metrical Theory for Optimal Continued Fractions, Journal of Number Theory 34 (1990), no. 3, 251–270.
• W. Bosma and C. Kraaikamp, Optimal approximation by continued fractions, J. Austral. Math. Soc. Ser. A50 (1991), no. 3, 481–504.
• W. Bosma, H. Jager and F. Wiedijk, Some metrical observations on the approximation by continued fractions, Nederl. Akad. Wetensch. Indag. Math. 45 (1983), no. 3, 281–299.
• F. Bagemihl and J. R. McLaughlin, Generalization of some classical theorems concerning triples of consecutive convergents to simple continued fractions, J. Reine Angew. Math. 221 (1966), 146–149.
• É. Borel, Contribution à l'analyse arithmétique du continu, Journal de Mathématiques Pures et Appliquées, 5e série, 9 (1903), 329–375.
• K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs 29, Mathematical Association of America, Washington, DC, 2002.
• H. Jager, Continued fractions and ergodic theory, transcendental numbers and related topics, RIMS Kōkyūroku 599 (1986), no. 1, 55–59.
• H. Jager and J. de Jonge, On the approximation by three consecutive continued fraction convergents, Indag. Math. (N.S.) 25 (2014), no. 4, 816–824.
• H. Jager and C. Kraaikamp, On the approximation by continued fractions, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), no. 3, 289–307.
• C. Kraaikamp, A new class of continued fraction expansions, Arith. 57 (1991), no. 1, 1–39.
• C. Kraaikamp, Statistic and ergodic properties of Minkowski's diagonal continued fraction, Theoret. Comput. Sci. 65 (1989), no. 2, 197–212.
• A. M. Legendre, Essay sur la théorie des nombres, Duprat, Paris, An VI, 1798.
• H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math. 4 (1981), no. 2, 399–426.
• O. Perron, Die Lehre von den Kettenbruchen, Chelsea, New York, 1929.
• G. J. Rieger, Mischung und Ergodizität bei Kettenbrchen nach nchsten Ganzen, J. Reine Angew. Math. 310 (1979), 171–181.
• A. M. Rockett, The metrical theory of continued fraction to the nearest integer, Acta Arith. 38 (1980), no. 2, 97–103.
• J. C. Tong, The conjugate property of the Borel theorem on Diophantine approximation, Math. Z. 184 (1983), no. 2, 151–153.
• J. C. Tong, Approximation by nearest integer continued fractions, II, Math. Scand. 74 (1994), no. 1, 17–18.
• K. Th. Vahlen, Ueber Näherungswerte und Kettenbrüche, J. Reine Angew. Math. 115 (1895), 221–233.
• H. C. Williams, On mid-period criteria for the nearest integer continued fraction expansion of $\sqrt{D}$, Utilitas Math. 27 (1985), 169–185.
• H. C. Williams and P. A. Buhr, Calculation of the regulator of Q($\sqrt{D}$) by use of the nearest integer continued fraction algorithm, Math. Comp. 33 (1979), no. 145, 369–381.