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Diophantine approximation of a single irrational number. (English) Zbl 0714.11036

For an irrational number x with simple continued fraction expansion \(x=[0;a_ 1,a_ 2,...]\) and (reduced) convergents \(p_ n/q_ n=[0;a_ 1,...,a_ n]\) let \(Q_ n=[a_ n,...,a_ 1]\), \(P_ n=[a_{n+2},a_{n+3},...]\). Rational approximation of x rests on the identity \(M_ n^{-1}=(-1)^ nq_ n(q_ nx-p_ n)\), where \(M_ n=a_{n+1}+P_ n^{-1}+Q_ n^{-1}\). Put \(f(Q,M)=Q+(M-Q^{-1})^{- 1}\). The following relations are evident: \(M_{n-1}=f(Q_ n,M_ n)\), \(M_{n+1}=f(P_ n,M_ n)\). The simple observation that f is an increasing function of Q, provided \(MQ>2\), leads to the following surprisingly useful implications, setting a foundation for the theory of diophantine approximation (symmetric or one-sided) of a single variable. Theorem: Let \(r,s\in {\mathbb{R}}_+\), \(r>a_{n+1}\), \(r-a_{n+1}-s^{- 1}\neq 0\). Put \(g_ n=(r-a_{n+1}-s^{-1})^{-1}+(a_{n+1}+s^{- 1})^{-1}\). Then \(M_ n\leq r\), \(P_ n<s\) imply \(M_{n-1}>g_ n\); \(M_ n\leq r\), \(Q_ n<s\) imply \(M_{n+1}>g_ n\); \(M_ n\geq r\), \(P_ n>s\) imply \(M_{n-1}<g_ n\); \(M_ n\geq r\), \(Q_ n>s\) imply \(M_{n+1}<g_ n\). Appropriate choice of r and s yields numerous known results widely spread in the literature.
Reviewer: G.Ramharter

MSC:

11J04 Homogeneous approximation to one number
11J70 Continued fractions and generalizations
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[1] Bagemihl, F.; McLaughlin, J. R., Generalization of some classical theorems concerning triples of consecutive convergents to simple continued fractions, J. Reine Angew. Math., 221, 146-149 (1966) · Zbl 0135.11104
[2] Borel, É., Contribution à l’analyse arithmétique du continu, J. Math. Pures Appl., 9, 329-375 (1903) · JFM 34.0239.01
[3] Hurwitz, A., Über die angenähert Darstellung der Irrationalzahlen durch rationale Brüche, Math. Ann., 39, 279-284 (1891) · JFM 23.0222.02
[4] Kopetzky, G.; Schnitzer, F. J., Bemerkungen zu einem Approximationssatz für regelmässige Kettenbrüche, J. Reine Angew. Math., 293/294, 437-440 (1977) · Zbl 0349.10023
[5] Kopetzky, G.; Schnitzer, F. J., A geometric approach to approximations by continued fractions, J. Austral. Math. Soc. Ser. A, 43, 176-186 (1987) · Zbl 0634.10030
[6] LeVeque, W. J., On asymmetric approximations, Michigan Math. J., 2, 1-6 (1953) · Zbl 0059.03901
[7] Lang, S., (Introduction to Diophantine Approximations (1966), Addison-Wesley: Addison-Wesley Reading, MA)
[8] Müller, M., Über die Approximation reeler Zahlen durch die Näherungsbrüche ihres regelmässigen Kettenbruches, Arch. Math., 6, 253-258 (1955) · Zbl 0064.04401
[9] Prasad, K. C.; Lari, M., A note on a theorem of Perron, (Proc. Amer. Math. Soc., 97 (1986)), 19-20 · Zbl 0593.10030
[10] Schmidt, W. M., Diophantine approximation, (Lecture Notes in Math., Vol. 785 (1980), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0529.10032
[11] Segre, B., Lattice points in infinite domains and asymmetric Diophantine approximation, Duke J. Math., 12, 337-365 (1945) · Zbl 0060.11807
[12] Tong, J., The conjugate property of the Borel thoerem on Diophantine approximation, Math. Z., 184, 151-153 (1983) · Zbl 0497.10024
[13] Tong, J., On two theorems of Kpetzky and Schnitzer on the approximation of continued fractions, J. Reine Angew. Math., 362, 1-3 (1985) · Zbl 0568.10019
[14] Tong, J., A theorem on approximation of irrational numbers by simple continued fractions, (Proc. Edingurgh Math. Soc., 31 (1988)), 197-204 · Zbl 0645.10008
[15] Tong, J., Segre’s theorem on asymmetric Diophantine approximation, J. Number Theory, 28, 116-118 (1988) · Zbl 0645.10009
[16] Tong, J., The conjugate property for Diophantine approximation of continued fractions, (Proc. Amer. Math. Soc., 105 (1989)) · Zbl 0663.10007
[17] Vahlen, T., Über Näherungswerte und Kettenbruche, J. Reine Angew. Math., 115, 221-233 (1895) · JFM 26.0230.01
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