Geodesic
The best Diophantine approximation functions by continued fractions
Mathematica Bohemica, Tome 121 (1996) no. 1, pp. 89-94

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Let $\xi=[a_0;a_1,a_2,\dots,a_i,\dots]$ be an irrational number in simple continued fraction expansion, $p_i/q_i=[a_0;a_1,a_2,\dots,a_i]$, $M_i=q_i^2 |\xi-p_i/q_i|$. In this note we find a function $G(R,r)$ such that \align{n+1}G(R,r), {n+1}>R\text{ and }M_{n-1}>r\text{ imply }M_n
Let $\xi=[a_0;a_1,a_2,\dots,a_i,\dots]$ be an irrational number in simple continued fraction expansion, $p_i/q_i=[a_0;a_1,a_2,\dots,a_i]$, $M_i=q_i^2 |\xi-p_i/q_i|$. In this note we find a function $G(R,r)$ such that \align{n+1}\text{ and }M_{n-1}\text{ imply }M_n>G(R,r), {n+1}>R\text{ and }M_{n-1}>r\text{ imply }M_n(R,r). \endalign Together with a result the author obtained, this shows that to find two best approximation functions $\tilde H(R,r)$ and $\tilde L(R,r)$ is a well-posed problem. This problem has not been solved yet.
DOI : 10.21136/MB.1996.125943
Classification : 11A55, 11J04, 11J70
Keywords: best diophantine approximation; continued fraction; diophantine approximation 
Tong, Jingcheng. The best Diophantine approximation functions by continued fractions. Mathematica Bohemica, Tome 121 (1996) no. 1, pp. 89-94. doi: 10.21136/MB.1996.125943
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[1] F. Bagemihl, 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. | MR | Zbl

[2] E. Borel: Contribution a l'analyse arithmetique du continu. J. Math. Pures Appl. 9 (1903), 329-375.

[3] M. Fujiwara: Bemerkung zur Theorie der Approximation der irrationalen Zahlen durch rationale Zahlen. Tôhoku Math. J. 14 (1918), 109-115.

[4] A. Hurwitz: Über die angenäherte Darstellung der irrationalzahlen durch rationale Bruche. Math. Ann. 39 (1891), 279-284. | DOI | MR

[5] W. J. LeVeque: On asymmetric approximations. Michigan Math. J. 2 (1953), 1-6. | DOI | MR

[6] M. Miller: Über die Approximation reeller Zahlen durch die Näherungsbruche ihres regelmässigen Kettenbruches. Arch. Math. (Basel) 6 (1955), 253-258. | DOI | MR

[7] M. B. Nathanson: Approximation by continued fractions. Proc. Amer. Math. Soc. 45 (1974), 323-324. | DOI | MR | Zbl

[8] O. Perron: Die Lehre von den Kettenbruchen I, II. 3rd ed. Teubner, Leipzig, 1954. | MR

[9] B. Segre: Lattice points in the infinite domains and asymmetric Diophantine approximation. Duke Math. J. 12 (1945), 337-365. | MR

[10] J. Tong: The conjugate property for Diophantine approximation of continued fraction. Proc. Amer. Math. Soc. 105 (1989), 535-539. | DOI | MR

[11] J. Tong: Diophantine approximation of a single irrational number. J. Number Theory 34 (1990), 53-57. | DOI | MR | Zbl

[12] J. Tong: Diophantine approximation by continued fractions. J. Austral. Math. Soc. Ser. A 51 (1991), 324-330. | DOI | MR | Zbl

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