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arXiv · 1908.06157

The Minkowski chain and Diophantine approximation

Nickolas Andersen·William Duke
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Abstract

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. We also give a variant of Dirichlet's approximation theorem for a real linear form that produces a whole basis of approximating integral vectors rather than a single one. This result holds if and only if the form is badly approximable. The proofs rely on properties of successive minima and reduced bases of lattices.

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Nickolas Andersen, William Duke. 2019-08-16. The Minkowski chain and Diophantine approximation. https://arxiv.org/abs/1908.06157

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