The conjugate property for Diophantine approximation of continued fractions
Author:
Jing Cheng Tong
Journal:
Proc. Amer. Math. Soc. 105 (1989), 535539
MSC:
Primary 11J04; Secondary 11J70, 11J72
MathSciNet review:
937852
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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 and at least one does not satisfy this inequality. (ii) Let be a positive real number. Among any four consecutive convergents , at least one satisfies and at least one does not satisfy this inequality, where if is odd, if is even.
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 [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), 146149. MR 0183999 (32:1475)
 [2]
 É. Borel, Contribution à l'analyse arithmétique du continu, J. Math. Pures Appl. (9) (1903), 329375.
 [3]
 W. J. LeVeque, On asymmetric approximations, Michigan Math. J. 2 (1953), 16. MR 0062784 (16:18b)
 [4]
 M. Müller, Über die Approximation reeler Zahlen durch die Näherungsbruche ihres regelmässigen Kettenbruches, Arch. Math. 6 (1955), 253258. MR 0069850 (16:1090e)
 [5]
 C. D. Olds, Note on an asymmetric Diophantine approximation, Bull. Amer. Math. Soc. 52 (1946), 261263. MR 0017758 (8:196e)
 [6]
 K. C. Prasad and M. Lari, A note on a theorem of Perron, Proc. Amer. Math. Soc. 97 (1986), 1920. MR 831378 (87h:11065)
 [7]
 B. Segre, Lattice points in infinite domains and asymmetric Diophantine approximation, Duke Math. J. 12 (1945), 337365. MR 0012096 (6:258a)
 [8]
 P. Szüsz, On a theorem of Segre, Acta Arith. 23 (1973), 371377. MR 0344202 (49:8942)
 [9]
 J. Tong, The conjugate property of the Borel theorem on Diophantine approximation, Math. Z. 184 (1983), 151153. MR 716268 (85m:11039)
 [10]
 , A theorem on approximation of irrational numbers by simple continued fractions, Proc. Edinburgh Math. Soc. 31 (1988), 197204. MR 989752 (90e:11101)
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 , Segre's theorem on asymmetric Diophantine approximation, J. Number Theory 28 (1988), 116118. MR 925611 (89h:11033)
 [12]
 , Symmetric and asymmetric Diophantine approximations of continued fractions, Bull. Soc. Math. France, (to appear). MR 1021563 (90k:11086)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198909378528
PII:
S 00029939(1989)09378528
Article copyright:
© Copyright 1989
American Mathematical Society
