# The Minkowski chain and Diophantine approximation

## TL;DR

This paper explores the Minkowski chain, a generalization of the Hurwitz chain, to establish criteria for algebraic numbers and real linear forms regarding their approximation properties, using lattice theory.

## Contribution

It introduces new criteria for badly approximable and singular forms based on Minkowski's generalization, extending classical Diophantine approximation results.

## Key findings

- Criteria for algebraic numbers using Minkowski chain
- A variant of Dirichlet's theorem producing a basis of approximations
- Characterization of badly approximable forms via lattice minima

## 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.