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Higher diophantine approximation exponents and continued fraction symmetries for function fields


Author: Dinesh S. Thakur
Journal: Proc. Amer. Math. Soc. 139 (2011), 11-19
MSC (2010): Primary 11J68, 11J70, 11J93
DOI: https://doi.org/10.1090/S0002-9939-2010-10584-1
Published electronically: August 18, 2010
MathSciNet review: 2729066
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Abstract: We construct many families of nonquadratic algebraic Laurent series with continued fractions having a bounded partial quotients sequence (the diophantine approximation exponent for approximation by rationals is thus $ 2$, agreeing with the Roth value) and with the diophantine approximation exponent for approximation by quadratics being arbitrarily large. In contrast, the Schmidt value (analog of the Roth value for approximations by quadratics, in the number field case) is $ 3$. We calculate diophantine approximation exponents for approximations by rationals for function field analogs of $ \pi$, $ e$ and Hurwitz numbers (which are transcendental) and also give an interesting lower bound (which may be the actual value) for the exponent for approximation by quadratics for the latter two. We do this exploiting the situation when `folding' or `negative reversal' patterns of the relevant continued fractions become `repeating' or `half-repeating' in even or odd characteristic respectively.


References [Enhancements On Off] (What's this?)

  • [BS76] L. Baum and M. Sweet, Continued fractions of algebraic power series in characteristic 2, Ann. of Math 103 (1976), 593-610. MR 0409372 (53:13127)
  • [BG06] E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Cambridge U. Press, Cambridge (2006). MR 2216774 (2007a:11092)
  • [B04] Y. Bugeaud, Approximation by algebraic numbers, Cambridge U. Press (2004). MR 2136100 (2006d:11085)
  • [KTV00] M. Kim, D. Thakur and J. F. Voloch, Diophantine approximation and deformation, Bull. Soc. Math. France 128 (2000), 585-598. MR 1815399 (2001k:11134)
  • [M49] K. Mahler, On a theorem of Liouville in fields of positive characteristic, Can. J. Math. 1 (1949), 397-400. MR 0031497 (11:159e)
  • [dM70] B. de Mathan, Approximations diophatiennes dans un corps local, Bull. Soc. Math. France Suppl. Mem. 21:93 (1970). MR 0274396 (43:161)
  • [MF73] M. Mendes France, Sur les continues fractions limitées, Acta Arith. 23 (1973), 207-215. MR 0323727 (48:2083)
  • [R03] D. Roy, Approximation to real numbers by cubic algebraic integers, Annals of Math. 158 (2003), 1081-1087. MR 2031862 (2004k:11110)
  • [Sa08] V. Kh. Salikhov, On the irrationality measure of $ \pi$, Communications of Moscow Math. Soc. 63 (2008) 570-572. MR 2483171 (2010b:11082)
  • [S80] W. Schmidt, Diophantine approximation, Lecture Notes in Math. 785, Springer Verlag, Berlin (1980). MR 568710 (81j:10038)
  • [S00] W. Schmidt, On continued fractions and diophantine approximation in power series fields, Acta Arith. XCV.2 (2000), 139-166. MR 1785412 (2001j:11063)
  • [T92] D. Thakur, Continued fraction for the exponential for $ F_q[t]$, J. Number Theory 41 (1992), 150-155. MR 1164793 (93d:11131)
  • [T96] D. Thakur, Exponential and continued fractions, J. Number Theory 59 (1996), 248-261. MR 1402608 (97h:11145)
  • [T97] D. Thakur, Patterns of continued fractions for the analogues of $ e$ and related numbers in the function field case, J. Number Theory 66 (1997), 129-147. MR 1467193 (98d:11145)
  • [T99] D. Thakur, Diophantine approximation exponents and continued fractions for algebraic power series, J. Number Theory 79 (1999), 284-291. MR 1728151 (2000j:11104)
  • [T04] D. Thakur, Function Field Arithmetic, World Scientific, NJ (2004). MR 2091265 (2005h:11115)
  • [T09] D. Thakur, Approximation exponents in function fields, Analytic Number Theory, Essays in honor of Klaus Roth, ed. by W. Chen, T. Gowers, H. Halberstam, W. Schmidt, R. Vaughn, Cambridge U. Press, Cambridge (2009), 421-435. MR 2508661 (2010b:11079)
  • [V88] J. F. Voloch, Diophantine approximation in finite characteristic, Period. Math. Hungar. 19 (1988), 217-225. MR 961018 (89h:11045)
  • [W] M. Waldschmidt, Report on some recent advances in Diophantine approximation, to be published (Springer) in special volume in honor of Serge Lang. Also at people.math.jussieu.fr/~miw/articles/pdf/miwLangeMemorialVolume.pdf

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Additional Information

Dinesh S. Thakur
Affiliation: Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721
Email: thakur@@math.arizona.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10584-1
Received by editor(s): January 2, 2010
Published electronically: August 18, 2010
Additional Notes: This research was supported in part by NSA grant H98230-08-1-0049
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2010 Dinesh S. Thakur

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