# On a theorem of Davenport and Schmidt

## TL;DR

This paper generalizes Davenport and Schmidt's work on improving Dirichlet's approximation theorems, using geometry of numbers and semi-regular continued fractions to establish sharp bounds in a generalized norm setting.

## Contribution

It introduces a new approach using semi-regular continued fractions to analyze approximation bounds in the geometry of numbers with arbitrary norms.

## Key findings

- Established sharp bounds for approximation improvements
- Extended classical results to general norms in $\\mathbb R^2$
- Utilized semi-regular continued fractions with a best approximation property

## Abstract

This work is motivated by a paper of Davenport and Schmidt, which treats the question of when Dirichlet's theorems on the rational approximation of one or of two irrationals can be improved and if so, by how much. We consider a generalization of this question in the simplest case of a single irrational but in the context of the geometry of numbers in $\mathbb R^2$, with the sup-norm replaced by a more general one. Results include sharp bounds for how much improvement is possible under various conditions. The proofs use semi-regular continued fractions that are characterized by a certain best approximation property determined by the norm.