Badly approximable systems of linear forms over a field of formal series
Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 421-444.

We prove that the Hausdorff dimension of the set of badly approximable systems of m linear forms in n 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.

Nous montrons que la dimension de Hausdorff de l’ensemble des systèmes mal approchables de m formes linéaires en n 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.

DOI: 10.5802/jtnb.552
Kristensen, Simon 1

1 Department of Mathematical Sciences Faculty of Science University of Aarhus Ny Munkegade, Building 530 8000 Aarhus C, Denmark
@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},
     mrnumber = {2289432},
     zbl = {05135397},
     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
DA  - 2006///
SP  - 421
EP  - 444
VL  - 18
IS  - 2
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.552/
UR  - https://www.ams.org/mathscinet-getitem?mr=2289432
UR  - https://zbmath.org/?q=an%3A05135397
UR  - https://doi.org/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 https://doi.org/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, Volume 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 p-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

Cited by Sources: