Nous montrons que la dimension de Hausdorff de l’ensemble des systèmes mal approchables de formes linéaires en variables sur le corps des séries de Laurent à coefficients dans un corps fini est maximale. Ce résultat est un analogue de la généralisation multidimensionnelle de Schmidt du théorème de Jarník sur les nombres mal approchables.
We prove that the Hausdorff dimension of the set of badly approximable systems of linear forms in variables over the field of Laurent series with coefficients from a finite field is maximal. This is an analogue of Schmidt’s multi-dimensional generalisation of Jarník’s Theorem on badly approximable numbers.
@article{JTNB_2006__18_2_421_0,
author = {Kristensen, Simon},
title = {Badly approximable systems of linear forms over a field of formal series},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {421--444},
publisher = {Universit\'e Bordeaux 1},
volume = {18},
number = {2},
year = {2006},
doi = {10.5802/jtnb.552},
zbl = {05135397},
mrnumber = {2289432},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.552/}
}
TY - JOUR AU - Kristensen, Simon TI - Badly approximable systems of linear forms over a field of formal series JO - Journal de théorie des nombres de Bordeaux PY - 2006 SP - 421 EP - 444 VL - 18 IS - 2 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.552/ DO - 10.5802/jtnb.552 LA - en ID - JTNB_2006__18_2_421_0 ER -
%0 Journal Article %A Kristensen, Simon %T Badly approximable systems of linear forms over a field of formal series %J Journal de théorie des nombres de Bordeaux %D 2006 %P 421-444 %V 18 %N 2 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.552/ %R 10.5802/jtnb.552 %G en %F JTNB_2006__18_2_421_0
Kristensen, Simon. Badly approximable systems of linear forms over a field of formal series. Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 421-444. doi : 10.5802/jtnb.552. http://www.numdam.org/articles/10.5802/jtnb.552/
[1] A. G. Abercrombie, Badly approximable -adic integers. Proc. Indian Acad. Sci. Math. Sci. 105 (2) (1995), 123–134. | MR | Zbl
[2] M. Amou, A metrical result on transcendence measures in certain fields. J. Number Theory 59 (2) (1996), 389–397. | MR | Zbl
[3] J. W. S. Cassels, An introduction to the geometry of numbers. Springer-Verlag, Berlin, 1997. | MR | Zbl
[4] V. Jarník, Zur metrischen Theorie der diophantischen Approximationen. Prace Mat.–Fiz. (1928–1929), 91–106.
[5] S. Kristensen, On the well-approximable matrices over a field of formal series. Math. Proc. Cambridge Philos. Soc. 135 (2) (2003), 255–268. | MR | Zbl
[6] S. Lang, Algebra (2nd Edition). Addison-Wesley Publishing Co., Reading, Mass., 1984. | MR | Zbl
[7] K. Mahler, An analogue to Minkowski’s geometry of numbers in a field of series. Ann. of Math. (2) 42 (1941), 488–522. | MR | Zbl
[8] H. Niederreiter, M. Vielhaber, Linear complexity profiles: Hausdorff dimensions for almost perfect profiles and measures for general profiles. J. Complexity 13 (3) (1997), 353–383. | MR | Zbl
[9] W. M. Schmidt, On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 123 (1966), 178–199. | MR | Zbl
[10] W. M. Schmidt, Badly approximable systems of linear forms, J. Number Theory 1 (1969), 139–154. | MR | Zbl
[11] V. G. Sprindžuk, Mahler’s problem in metric number theory. American Mathematical Society, Providence, R.I., 1969. | MR | Zbl
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