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A theorem on approximation of irrational numbers by simple continued fractions

Published online by Cambridge University Press:  20 January 2009

Jingcheng Tong
Jingcheng Tong
Affiliation:
Department of Mathematical Sciences, University of North Florida, Jacksonville, FL 32216, U.S.A. Institute of Applied Mathematics, Academia Sinica, Peking, China
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Let ξ be an irrational number with simple continued fraction expansion ξ= [a0;a1,a2,…], Pn/qn be its nth convergent, . The following two theorems were proved by Müller [9] and rediscovered by Bagemihl and McLaughlin [1]:

Theorem 1.For n>1,

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 31 , Issue 2 , June 1988 , pp. 197 - 204
Copyright
Copyright © Edinburgh Mathematical Society 1988

References

REFERENCES

1.Bagemihl, F. and McLaughlin, J. R., Generalization of some classical theorems concerning triples of consecutive convergents to simple continued fractions, J. Reine Angew. Math. 221 (1966), 146–149.Google Scholar
2.Borel, E., Contribution à l'analyse arithmetique du continu, J. Math. Pures Appl. 9 (1903), 329–375.Google Scholar
3.Cohn, J. H. E., Hurwitz' theorem, Proc. Amer. Math. Soc. 38 (1973), 436.Google Scholar
4.Fujiwara, M., Bemerkung zur Theorie der Approximation der irrationalen Zahlen durch rationale Zahlen, Tohoku Math. J. 14 (1918), 109–115.Google Scholar
5.Koksma, J. F., Diophantische Approximationen (Chelsea, New York, 1936).Google Scholar
6.Kurosu, K., Note on the theory of approximation of irrational numbers by rational numbers, Tohoku Math. J. 21 (1922), 247–260.Google Scholar
7.Lang, S., Introduction to Diophantine Approximations, (Addison-Wesley Publ. Co., 1966).Google Scholar
8.LeVeque, W. J., On asymmetric approximations, Michigan Math. J. 2 (1953), 1–6.CrossRefGoogle Scholar
9.Müller, M., Über die Approximation reeler Zahlen durch die Näherungsbruche ihres regelmässigen Kettenbruches, Arch. Math. 6 (1955), 253–258.CrossRefGoogle Scholar
10.Nathanson, M. B., Approximation by continued fractions, Proc. Amer. Math. Soc. 45 (1974), 323–324.CrossRefGoogle Scholar
11.Niven, I., On asymmetric Diophantine approximations, Michigan Math. J. 9 (1962), 121–123.CrossRefGoogle Scholar
12.Niven, I., Diophantine Approximations (Interscience Publishers, 1963).Google Scholar
13.Olds, C. D., Note on an asymmetric Diophantine approximation, Bull. Amer. Math. Soc. 52 (1946), 261–263.CrossRefGoogle Scholar
14.Robinson, R. M., Unsymmetric approximation of irrational numbers, Bull. Amer. Math. Soc. 53 (1947), 351–361.CrossRefGoogle Scholar
15.Robinson, R. M., The critical numbers for unsymmetric approximation, Bull. Amer. Math. Soc. 54 (1948), 693–705.CrossRefGoogle Scholar
16.Schmidt, A. L., Approximation theorems of Borel and Fujiwara, Math. Scand. 14 (1964), 35–38.CrossRefGoogle Scholar
17.Schmidt, W., Diophantine Approximation (Lecture Notes in Math. 785, Springer-Verlag, 1980).Google Scholar
18.Segre, B., Lattice points in infinite domains and asymmetric Diophantine approximation, Duke J. Math. 12 (1945), 337–365.CrossRefGoogle Scholar
19.Szüsz, P., On a theorem of Segre, Acta Arith. 23 (1973), 371–377.CrossRefGoogle Scholar
20.Tong, J., A generalization of the Borel theorem on Diophantine approximation, Riv. Math. Univ. Parma 9 (1983), 121–124.Google Scholar
21.Tong, J., The conjugate property of the Borel theorem on Diophantine approximation, Math. Z. 184 (1983), 151–153.CrossRefGoogle Scholar
22.Vahlen, K. T., Über Näherungswerte und Kettenbruche, J. Reine Angew. Math. 115 (1895), 221–233.Google Scholar