Remarks on Diophantine approximation in function fields

  • Arijit Ganguly
  • Anish Ghosh


We study some problems in metric Diophantine approximation over local fields of positive characteristic.


Beresnevich, V., Badly approximable points on manifolds, Invent. Math. 202 (2015), no. 3, 1199–1240.

Bugeaud, Y., Approximation by algebraic numbers, Cambridge Tracts in Mathematics, no. 160, Cambridge University Press, Cambridge, 2004.

Bundschuh, P., Transzendenzmasse in Körpern formaler Laurentreihen, J. Reine Angew. Math. 299/300 (1978), 411–432.

Cassels, J. W. S., An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, no. 45, Cambridge University Press, New York, 1957.

Davenport, H. and Schmidt, W. M., Dirichlet's theorem on diophantine approximation. II, Acta Arith. 16 (1969/1970), 413–424.

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.

Dubois, E., On Mahler's classification in Laurent series fields, Rocky Mountain J. Math. 26 (1996), no. 3, 1003–1016.

Ganguly, A. and Ghosh, A., Dirichlet's theorem in function fields, Canad. J. Math. 69 (2017), no. 3, 532–547.

Ghosh, A., Metric Diophantine approximation over a local field of positive characteristic, J. Number Theory 124 (2007), no. 2, 454–469.

Khintchine, A., Über eine Klasse linearer diophantischer Approximationen, Rend. Circ. Mat. Palermo 50 (1926), 170–195.

Kleinbock, D. Y., Lindenstrauss, E., and Weiss, B., On fractal measures and Diophantine approximation, Selecta Math. (N.S.) 10 (2004), no. 4, 479–523.

Kleinbock, D. Y. and Margulis, G. A., Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2) 148 (1998), no. 1, 339–360.

Kleinbock, D. Y. and Tomanov, G., Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv. 82 (2007), no. 3, 519–581.

Kristensen, S., Pedersen, S. H., and Weiss, B., Some remarks on Mahler's classification in higher dimension, Mosc. J. Comb. Number Theory 6 (2016), no. 2-3, 177–190.

Lasjaunias, A., A survey of Diophantine approximation in fields of power series, Monatsh. Math. 130 (2000), no. 3, 211–229.

Lasjaunias, A., Diophantine approximation and continued fractions in power series fields, in “Analytic number theory”, Cambridge Univ. Press, Cambridge, 2009, pp. 297--305.

Mahler, K., Über das Maßder Menge aller $S$-Zahlen, Math. Ann. 106 (1932), no. 1, 131–139.

Mahler, K., Zur Approximation der Exponentialfunktion und des Logarithmus. Teil I, J. Reine Angew. Math. 166 (1932), 118–136.

Mahler, K., An analogue to Minkowski's geometry of numbers in a field of series, Ann. of Math. (2) 42 (1941), 488–522.

Mahler, K., On a theorem of Liouville in fields of positive characteristic, Canadian J. Math. 1 (1949), 397–400.

de Mathan, B., Approximations diophantiennes dans un corps local, Bull. Soc. Math. France Suppl. Mém. 21 (1970), 93 pp.

Ooto, T., On Diophantine exponents for Laurent series over a finite field, J. Number Theory 185 (2018), 349–378.

Sprindžuk, V. G., Mahler's problem in metric number theory, Translations of Mathematical Monographs, no. 25, American Mathematical Society, Providence, R.I., 1969.

Sprindžuk, V. G., Achievements and problems of the theory of Diophantine approximations, Uspekhi Mat. Nauk 35 (1980), no. 4(214), 3–68.

Yu, K. R., A generalization of Mahler's classification to several variables, J. Reine Angew. Math. 377 (1987), 113–126.
How to Cite
Ganguly, A., & Ghosh, A. (2019). Remarks on Diophantine approximation in function fields. MATHEMATICA SCANDINAVICA, 124(1), 5-14.