Remarks on Diophantine approximation in function fields

  • Arijit Ganguly
  • Anish Ghosh

Abstract

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

References

Beresnevich, V., Badly approximable points on manifolds, Invent. Math. 202 (2015), no. 3, 1199–1240. https://doi.org/10.1007/s00222-015-0586-8

Bugeaud, Y., Approximation by algebraic numbers, Cambridge Tracts in Mathematics, no. 160, Cambridge University Press, Cambridge, 2004. https://doi.org/10.1017/CBO9780511542886

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. https://doi.org/10.4064/aa-16-4-413-424

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. https://doi.org/10.1216/rmjm/1181072033

Ganguly, A. and Ghosh, A., Dirichlet's theorem in function fields, Canad. J. Math. 69 (2017), no. 3, 532–547. https://doi.org/10.4153/CJM-2016-024-2

Ghosh, A., Metric Diophantine approximation over a local field of positive characteristic, J. Number Theory 124 (2007), no. 2, 454–469. https://doi.org/10.1016/j.jnt.2006.10.009

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. https://doi.org/10.1007/s00029-004-0378-2

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. https://doi.org/10.2307/120997

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. https://doi.org/10.4171/CMH/102

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. https://doi.org/10.1007/s006050070036

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. https://doi.org/10.1007/BF01455882

Mahler, K., Zur Approximation der Exponentialfunktion und des Logarithmus. Teil I, J. Reine Angew. Math. 166 (1932), 118–136. https://doi.org/10.1515/crll.1932.166.118

Mahler, K., An analogue to Minkowski's geometry of numbers in a field of series, Ann. of Math. (2) 42 (1941), 488–522. https://doi.org/10.2307/1968914

Mahler, K., On a theorem of Liouville in fields of positive characteristic, Canadian J. Math. 1 (1949), 397–400. https://doi.org/10.4153/cjm-1949-035-0

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. https://doi.org/10.1016/j.jnt.2017.09.008

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. https://doi.org/10.1515/crll.1987.377.113
Published
2019-01-13
How to Cite
Ganguly, A., & Ghosh, A. (2019). Remarks on Diophantine approximation in function fields. MATHEMATICA SCANDINAVICA, 124(1), 5-14. https://doi.org/10.7146/math.scand.a-109985
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Articles