An analogue of continued fractions in number theory for Nevanlinna theory
Author:
Zhuan Ye
Journal:
Trans. Amer. Math. Soc. 356 (2004), 4829-4838
MSC (2000):
Primary 30D35, 11J70
DOI:
https://doi.org/10.1090/S0002-9947-04-03709-2
Published electronically:
June 25, 2004
MathSciNet review:
2084400
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We show an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory.
- Huaihui Chen and Zhuan Ye, The error term in Nevanlinna’s inequality, Sci. China Ser. A 43 (2000), no. 10, 1060–1066. MR 1802149, DOI https://doi.org/10.1007/BF02898240
- William Cherry and Zhuan Ye, Nevanlinna’s theory of value distribution, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. The second main theorem and its error terms. MR 1831783
- W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
- A. Hinkkanen, A sharp form of Nevanlinna’s second fundamental theorem, Invent. Math. 108 (1992), no. 3, 549–574. MR 1163238, DOI https://doi.org/10.1007/BF02100617
- A. Ya. Khinchin, Continued fractions, The University of Chicago Press, Chicago, Ill.-London, 1964. MR 0161833
- Serge Lang, The error term in Nevanlinna theory, Duke Math. J. 56 (1988), no. 1, 193–218. MR 932862, DOI https://doi.org/10.1215/S0012-7094-88-05609-8
- Serge Lang, The error term in Nevanlinna theory. II, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 115–125. MR 1003864, DOI https://doi.org/10.1090/S0273-0979-1990-15857-4
- Serge Lang, Introduction to Diophantine approximations, 2nd ed., Springer-Verlag, New York, 1995. MR 1348400
- Serge Lang and William Cherry, Topics in Nevanlinna theory, Lecture Notes in Mathematics, vol. 1433, Springer-Verlag, Berlin, 1990. With an appendix by Zhuan Ye. MR 1069755
- Serge Lang and Hale Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972). MR 306131, DOI https://doi.org/10.1007/978-1-4612-2120-3_5
- Joseph Miles, A sharp form of the lemma on the logarithmic derivative, J. London Math. Soc. (2) 45 (1992), no. 2, 243–254. MR 1171552, DOI https://doi.org/10.1112/jlms/s2-45.2.243
- Rolf Nevanlinna, Analytic functions, Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. Translated from the second German edition by Phillip Emig. MR 0279280
- Charles F. Osgood, Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better, J. Number Theory 21 (1985), no. 3, 347–389. MR 814011, DOI https://doi.org/10.1016/0022-314X%2885%2990061-7
- Min Ru, Nevanlinna theory and its relation to Diophantine approximation, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1850002
- Min Ru and Paul Vojta, Schmidt’s subspace theorem with moving targets, Invent. Math. 127 (1997), no. 1, 51–65. MR 1423025, DOI https://doi.org/10.1007/s002220050114
- L. R. Sons and Zhuan Ye, The best error terms of classical functions, Complex Variables Theory Appl. 28 (1995), no. 1, 55–66. MR 1700264
- Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451
- Paul Vojta, Nevanlinna theory and Diophantine approximation, Several complex variables (Berkeley, CA, 1995–1996) Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, Cambridge, 1999, pp. 535–564. MR 1748613
- Yuefei Wang, Sharp forms of Nevanlinna’s error terms, J. Anal. Math. 71 (1997), 87–102. MR 1454245, DOI https://doi.org/10.1007/BF02788024
- Pit-Mann Wong and Wilhelm Stoll, Second main theorem of Nevanlinna theory for nonequidimensional meromorphic maps, Amer. J. Math. 116 (1994), no. 5, 1031–1071. MR 1296724, DOI https://doi.org/10.2307/2374940
- Zhuan Ye, On Nevanlinna’s error terms, Duke Math. J. 64 (1991), no. 2, 243–260. MR 1136375, DOI https://doi.org/10.1215/S0012-7094-91-06412-4
- Zhuan Ye, On Nevanlinna’s second main theorem in projective space, Invent. Math. 122 (1995), no. 3, 475–507. MR 1359601, DOI https://doi.org/10.1007/BF01231453
- Zhuan Ye, An analogue of Khinchin’s theorem in Diophantine approximation for Nevanlinna theory, XVIth Rolf Nevanlinna Colloquium (Joensuu, 1995) de Gruyter, Berlin, 1996, pp. 309–319. MR 1427096, DOI https://doi.org/10.1002/%28SICI%291099-0488%2819960130%2934%3A2%3C309%3A%3AAID-POLB11%3E3.0.CO%3B2-N
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Additional Information
Zhuan Ye
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115
Email:
ye@math.niu.edu
Keywords:
Continued fraction,
meromorphic function,
approximation
Received by editor(s):
July 25, 2002
Published electronically:
June 25, 2004
Article copyright:
© Copyright 2004
American Mathematical Society