# \"{U}ber die Winkel zwischen Unterr\"{a}umen

## TL;DR

This paper investigates how well rational subspaces approximate real subspaces in Euclidean space, providing partial solutions to a longstanding problem about the relationship between approximation quality and the height of rational subspaces.

## Contribution

It offers a metric result on the approximation of linear subspaces by rational subspaces with bounded height, addressing a problem posed by Schmidt in 1967, especially in the case of 4-dimensional space.

## Key findings

- Partial solution for the case d=4, n=2
- Establishes bounds on the angle of approximation
- Advances understanding of rational subspace approximation

## Abstract

We prove a metric statement about approximation of a $n$-dimensional linear subspace $A$ in $\mathbb{R}^d$ by $n$-dimensional rational subspaces. We consider the problem of finding a rational subspace $B$ of bounded height $H=H(B)$ for which the angle of inclination $\psi (A,B) $ is small in terms of $H$. In the simplest case $d=4, n=2$ we give a partial solution of a problem formulated by W.M. Schmidt in 1967.