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Journal de Th´

eorie des Nombres

de Bordeaux

18

_{(2006), 421–444}

Badly approximable systems of linear forms over

a field of formal series

par

Simon KRISTENSEN

R´esum´e. _{Nous montrons que la dimension de Hausdorff de} l’en-semble des syst`emes mal approchables dem formes lin´eaires en nvariables sur le corps des s´eries de Laurent `a coefficients dans

un corps fini est maximale. Ce r´esultat est un analogue de la g´en´eralisation multidimensionnelle de Schmidt du th´eor`eme de Jarn´ık sur les nombres mal approchables.

Abstract. _{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 Jarn´ık’s Theorem on badly approximable num-bers.

SimonKristensen

Department of Mathematical Sciences Faculty of Science

University of Aarhus Ny Munkegade, Building 530 8000 Aarhus C, Denmark

E-mail:sik@imf.au.dk