Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T14:07:32.625Z Has data issue: false hasContentIssue false
  • English
  • Français

Dirichlet's Theorem in Function Fields

Published online by Cambridge University Press:  20 November 2018

Arijit Ganguly  and
Anish Ghosh
Arijit Ganguly
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 400005, India e-mail: [email protected], [email protected]
Anish Ghosh
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 400005, India e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study metric Diophantine approximation for function fields, specifically, the problem of improving Dirichlet's theorem in Diophantine approximation.

Keywords

11J8311K6037D4037A1722E40Dirichlet's theoremDiophantine approximationpositive characteristic
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 3 , 01 June 2017 , pp. 532 - 547
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Adamczewski, B. and Bugeaud, Y., On the Littlewood conjecture infields of power series. In: Probability and number theory-Kanazawa 2005, Adv. Stud. Pure Math.,49, Math. Soc. Japan, Tokyo, 2007, pp. 1–20.Google Scholar
[2] Amou, M., A metrical result on transcendence measures in certain fields. J. Number Theory 59(1996), no. 2, 389–397.http://dx.doi.org/10.1006/jnth.1996.0104 Google Scholar
[3] Athreya, J. S., Ghosh, A., and Prasad, A., Ultrametric logarithm laws, II. Monatsh. Math. 167(2012), no. 3-4, 333–356.http://dx.doi.Org/10.1007/s00605-012-0376-y Google Scholar
[4] Baker, R. C., Metric diophantine approximation on manifolds. J. London Math. Soc. (2) 14(1976), 43–48.http://dx.doi.Org/10.1112/jlms/s2-14.1.43 Google Scholar
[5] Baker, R. C., Dirichlet-s theorem on Diophantine approximation. Math. Proc. Cambridge Phil. Soc. 83(1978), no. 1, 37–59.http://dx.doi.org/10.1017/S030500410005427X Google Scholar
[6] Bugeaud, Y., Approximation by algebraic integers and Hausdorff dimension. J. London Math. Soc. (2) 65(2002), no. 3, 547–559. http://dx.doi.Org/10.1112/S0024610702003137 Google Scholar
[7] Davenport, H. and Schmidt, W. M., Dirichlet-s theorem on diophantine approximation. In: Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 113–132.Google Scholar
[8] Davenport, H., Dirichlet-s theorem on diophantine approximation. II.Acta Arith. 16(1969/1970), 413–424.Google Scholar
[9] de Mathan, B., Approximations diophantiennes dans un corps local. Bull. Soc. Math. France Suppl. Mé m. 21(1970), 1–93.Google Scholar
[10] Dodson, M., Rynne, B., and Vickers, J., Dirichlet-s theorem and Diophantine approximation on manifolds. J. Number Theory 36(1990), no. 1, 85–88.http://dx.doi.Org/10.1016/0022-314X(90)90006-D Google Scholar
[11] Dodson, M. M., Kristensen, S., and Levesley, J., A quantitative Khintchine- Groshev type theorem over afield of formal series. Indag. Math. (N.S.) 16(2005), no. 2,171–177.http://dx.doi.org/10.1016/S0019-3577(05)80020-5 Google Scholar
[12] Ganguly, A., Problems in Diophantine approximation and dynamical systems. PhD thesis, Tata Institute of Fundamental Research, forthcoming.Google Scholar
[13] Ghosh, A., Metric Diophantine approximation over a local field of positive characteristic. J. Number Theory 124(2007), no. 2, 454–469.http://dx.doi.Org/10.1016/j.jnt.2006.10.009 Google Scholar
[14] Ghosh, A. and Royals, R., An extension of Khintchine's theorem. Acta Arith. 167(2015), no. 1,1–17.http://dx.doi.org/10.4064/aa167-1-1 Google Scholar
[15] Kleinbock, D. Y. and Margulis, G. A., Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. 148(1998), no. 1, 339–360.http://dx.doi.Org/10.2307/120997 Google Scholar
[16] Kleinbock, D.,Lindenstrauss, E., and Weiss, B., On fractal measures and Diophantine approximation. Selecta Math. 10(2004), no. 4, 479–523. http://dx.doi.org/10.1007/s00029-004-0378-2 Google Scholar
[17] Kleinbock, D. and Tomanov, G., Flows on S-larithmetic homogeneous spaces and applications to metric Diophantine approximation. Comment. Math. Helv. 82(2007), no. 3, 519–581.http://dx.doi.Org/10.4171/CMH/102 Google Scholar
[18] Kleinbock, D. and Weiss, B., Dirichlet-s theorem on diophantine approximation and homogeneous flows. J. Mod. Dyn. 2(2008), no. 1, 43–62.Google Scholar
[19] Kleinbock, D., Friendly measures, homogeneous flows and singular vectors. In: Algebraic and topological dynamics, Contemp. Math., 385, Amer. Math. Soc, Providence, RI, 2005, pp. 281–292. http://dx.doi.Org/10.1090/conm/385/07201 Google Scholar
[20] Kristensen, S., On well approximable matrices over afield of formal series. Math. Proc. Cambridge Philos. Soc. 135(2003), no. 2, 255–268.http://dx.doi.Org/10.1017/S0305004103006911 Google Scholar
[21] Kristensen, S., Badly approximable systems of linear forms over afield of formal series. J. Théor.Nombres Bordeaux 18(2006), no. 2, 421–444. http://dx.doi.Org/10.58O2/jtnb.552 Google Scholar
[22] Lasjaunias, A., A survey of Diophantine approximation infields of power series. Monatsh. Math. 130(2000), no. 3, 211–229.http://dx.doi.Org/10.1007/s006050070036 Google Scholar
[23] Lasjaunias, A., Diophantine approximation and continued fractions in power series fields. In: Analytic number theory, Cambridge Univ. Press, Cambridge, 2009, pp. 297–305.Google Scholar
[24] Mahler, K., An analogue to Minkowski's geometry of numbers in afield of series. Ann. of Math. (2) 42(1941), no. 2, 488–522.http://dx.doi.Org/10.2307/1968914 Google Scholar
[25] Mattila, P., Geometry of sets and measures in Euclidean space. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995.http://dx.doi.Org/10.1017/CBO9780511623813 Google Scholar
[26] Miller, B., The existence of measures of a given cocycle, I: atomless, ergodic σ-finite measures. Ergodic Theory Dynam. System 28(2008), no. 5,1599–1613.http://dx.doi.Org/10.1017/S0143385707001113 Google Scholar
[27] Schikhof, W. H., Ultrametric calculus. An introduction to p-adic analysis. Cambridge Studies in Advanced Mathematics, 4, Cambridge University Press, Cambridge, 1984.Google Scholar
[28] Shah, N. A., Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation. Invent. Math. 177(2009), no. 3, 509–532.http://dx.doi.org/10.1007/s00222-009-0186-6 Google Scholar
[29] Shah, N. A., Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms. J. Amer. Math. Soc. 23(2010), no. 2, 563–589.http://dx.doi.Org/10.1090/S0894-0347-09-00657-2 Google Scholar
[30] Shah, N. A., Equidistribution of translates of curves on homogeneous spaces and Dirichlet's approximation. Proceedings of the International Congress of Mathematicians, Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1332–1343.Google Scholar
[31] Sprindzuk, V. G., Mahler's problem in metric number theory. Translations of Mathematical Monographs, 25, American Mathematical Society, Providence, RI, 1969.Google Scholar
[32] Sprindzuk, V. G., Achievements and problems in Diophantine approximations. (Russian) Uspekhi Mat. Nauk 35(1980), no. 4(214), 3–68, 248.Google Scholar