The conjugate property for Diophantine approximation of continued fractions

Author:
Jing Cheng Tong

Journal:
Proc. Amer. Math. Soc. **105** (1989), 535-539

MSC:
Primary 11J04; Secondary 11J70, 11J72

DOI:
https://doi.org/10.1090/S0002-9939-1989-0937852-8

MathSciNet review:
937852

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an irrational number with simple continued fraction expansion , and be its th convergent. In this paper we first prove the duality of some inequalities, and then prove the following conjugate properties for symmetric and asymmetric Diophantine approximations.

(i) Among any three consecutive convergents , at least one satisfies

(ii) Let be a positive real number. Among any four consecutive convergents , at least one satisfies

**[1]**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. MR**0183999****[2]**É. Borel,*Contribution à l'analyse arithmétique du continu*, J. Math. Pures Appl. (9) (1903), 329-375.**[3]**W. J. LeVeque,*On asymmetric approximations*, Michigan Math. J.**2**(1954), 1–6. MR**0062784****[4]**Max Müller,*Über die Approximation reeller Zahlen durch die Näherungsbrüche ihres regelmässigen Kettenbruches*, Arch. Math.**6**(1955), 253–258 (German). MR**0069850**, https://doi.org/10.1007/BF01899402**[5]**C. D. Olds,*Note on an asymmetric Diophantine approximation*, Bull. Amer. Math. Soc.**52**(1946), 261–263. MR**0017758**, https://doi.org/10.1090/S0002-9904-1946-08554-7**[6]**K. C. Prasad and M. Lari,*A note on a theorem of Perron*, Proc. Amer. Math. Soc.**97**(1986), no. 1, 19–20. MR**831378**, https://doi.org/10.1090/S0002-9939-1986-0831378-5**[7]**B. Segre,*Lattice points in infinite domains and asymmetric Diophantine approximations*, Duke Math. J.**12**(1945), 337–365. MR**0012096****[8]**P. Szüsz,*On a theorem of Segre*, Acta Arith.**23**(1973), 371–377. MR**0344202**, https://doi.org/10.4064/aa-23-4-371-377**[9]**Jing Cheng Tong,*The conjugate property of the Borel theorem on Diophantine approximation*, Math. Z.**184**(1983), no. 2, 151–153. MR**716268**, https://doi.org/10.1007/BF01252854**[10]**Jing Cheng Tong,*A theorem on approximation of irrational numbers by simple continued fractions*, Proc. Edinburgh Math. Soc. (2)**31**(1988), no. 2, 197–204. MR**989752**, https://doi.org/10.1017/S001309150000331X**[11]**Jing Cheng Tong,*Segre’s theorem on asymmetric Diophantine approximation*, J. Number Theory**28**(1988), no. 1, 116–118. MR**925611**, https://doi.org/10.1016/0022-314X(88)90122-9**[12]**Jing Cheng Tong,*Symmetric and asymmetric Diophantine approximation of continued fractions*, Bull. Soc. Math. France**117**(1989), no. 1, 59–67 (English, with French summary). MR**1021563**

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DOI:
https://doi.org/10.1090/S0002-9939-1989-0937852-8

Article copyright:
© Copyright 1989
American Mathematical Society