The Minkowski chain and Diophantine approximation

Andersen, Nickolas; Duke, William

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 503-523.

La chaîne de Hurwitz donne une suite de paires d’approximations de Farey d’un nombre réel irrationnel. Minkowski a donné un critère d’algébraicité d’un nombre en utilisant une certaine généralisation de la chaîne de Hurwitz. Nous appliquons cette généralisation (la chaîne de Minkowski) pour donner des critères pour qu’une forme linéaire réelle soit mal approchable ou singulière. Les preuves reposent sur des propriétés des minima successifs et des bases réduites de réseaux.

The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski’s generalization (the Minkowski chain) to give criteria for a real linear form to be either badly approximable or singular. The proofs rely on properties of successive minima and reduced bases of lattices.

Classification : 11J13, 11J70
Mots clés : Minkowski chain, Diophantine approximation

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