Skip to main content Accessibility help
×
Home

Dirichlet's Theorem in Function Fields

  • Arijit Ganguly (a1) and Anish Ghosh (a1)

Abstract

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

Copyright

References

Hide All
[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. 120.
[2] Amou, M., A metrical result on transcendence measures in certain fields. J. Number Theory 59(1996), no. 2, 389397.http://dx.doi.org/10.1006/jnth.1996.0104
[3] Athreya, J. S., Ghosh, A., and Prasad, A., Ultrametric logarithm laws, II. Monatsh. Math. 167(2012), no. 3-4, 333356.http://dx.doi.Org/10.1007/s00605-012-0376-y
[4] Baker, R. C., Metric diophantine approximation on manifolds. J. London Math. Soc. (2) 14(1976), 4348.http://dx.doi.Org/10.1112/jlms/s2-14.1.43
[5] Baker, R. C., Dirichlet-s theorem on Diophantine approximation. Math. Proc. Cambridge Phil. Soc. 83(1978), no. 1, 3759.http://dx.doi.org/10.1017/S030500410005427X
[6] Bugeaud, Y., Approximation by algebraic integers and Hausdorff dimension. J. London Math. Soc. (2) 65(2002), no. 3, 547559. http://dx.doi.Org/10.1112/S0024610702003137
[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. 113132.
[8] Davenport, H., Dirichlet-s theorem on diophantine approximation. II.Acta Arith. 16(1969/1970), 413424.
[9] de Mathan, B., Approximations diophantiennes dans un corps local. Bull. Soc. Math. France Suppl. Mé m. 21(1970), 193.
[10] Dodson, M., Rynne, B., and Vickers, J., Dirichlet-s theorem and Diophantine approximation on manifolds. J. Number Theory 36(1990), no. 1, 8588.http://dx.doi.Org/10.1016/0022-314X(90)90006-D
[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. 2,171177.http://dx.doi.org/10.1016/S0019-3577(05)80020-5
[12] Ganguly, A., Problems in Diophantine approximation and dynamical systems. PhD thesis, Tata Institute of Fundamental Research, forthcoming.
[13] Ghosh, A., Metric Diophantine approximation over a local field of positive characteristic. J. Number Theory 124(2007), no. 2, 454469.http://dx.doi.Org/10.1016/j.jnt.2006.10.009
[14] Ghosh, A. and Royals, R., An extension of Khintchine's theorem. Acta Arith. 167(2015), no. 1,117.http://dx.doi.org/10.4064/aa167-1-1
[15] Kleinbock, D. Y. and Margulis, G. A., Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. 148(1998), no. 1, 339360.http://dx.doi.Org/10.2307/120997
[16] Kleinbock, D.,Lindenstrauss, E., and Weiss, B., On fractal measures and Diophantine approximation. Selecta Math. 10(2004), no. 4, 479523. http://dx.doi.org/10.1007/s00029-004-0378-2
[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, 519581.http://dx.doi.Org/10.4171/CMH/102
[18] Kleinbock, D. and Weiss, B., Dirichlet-s theorem on diophantine approximation and homogeneous flows. J. Mod. Dyn. 2(2008), no. 1, 4362.
[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. 281292. http://dx.doi.Org/10.1090/conm/385/07201
[20] Kristensen, S., On well approximable matrices over afield of formal series. Math. Proc. Cambridge Philos. Soc. 135(2003), no. 2, 255268.http://dx.doi.Org/10.1017/S0305004103006911
[21] Kristensen, S., Badly approximable systems of linear forms over afield of formal series. J. Théor.Nombres Bordeaux 18(2006), no. 2, 421444. http://dx.doi.Org/10.58O2/jtnb.552
[22] Lasjaunias, A., A survey of Diophantine approximation infields of power series. Monatsh. Math. 130(2000), no. 3, 211229.http://dx.doi.Org/10.1007/s006050070036
[23] Lasjaunias, A., Diophantine approximation and continued fractions in power series fields. In: Analytic number theory, Cambridge Univ. Press, Cambridge, 2009, pp. 297305.
[24] Mahler, K., An analogue to Minkowski's geometry of numbers in afield of series. Ann. of Math. (2) 42(1941), no. 2, 488522.http://dx.doi.Org/10.2307/1968914
[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
[26] Miller, B., The existence of measures of a given cocycle, I: atomless, ergodic σ-finite measures. Ergodic Theory Dynam. System 28(2008), no. 5,15991613.http://dx.doi.Org/10.1017/S0143385707001113
[27] Schikhof, W. H., Ultrametric calculus. An introduction to p-adic analysis. Cambridge Studies in Advanced Mathematics, 4, Cambridge University Press, Cambridge, 1984.
[28] Shah, N. A., Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation. Invent. Math. 177(2009), no. 3, 509532.http://dx.doi.org/10.1007/s00222-009-0186-6
[29] Shah, N. A., Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms. J. Amer. Math. Soc. 23(2010), no. 2, 563589.http://dx.doi.Org/10.1090/S0894-0347-09-00657-2
[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. 13321343.
[31] Sprindzuk, V. G., Mahler's problem in metric number theory. Translations of Mathematical Monographs, 25, American Mathematical Society, Providence, RI, 1969.
[32] Sprindzuk, V. G., Achievements and problems in Diophantine approximations. (Russian) Uspekhi Mat. Nauk 35(1980), no. 4(214), 368, 248.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Related content

Powered by UNSILO

Dirichlet's Theorem in Function Fields

  • Arijit Ganguly (a1) and Anish Ghosh (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.