On the approximation exponents for subspaces of ${\mathbb{R}}^n$

Abstract

This paper follows the generalisation of the classical theory of Diophantine approximation to subspaces of ${\mathbb{R}}^n$ established by W. M. Schmidt in 1967. Let $A$ and $B$ be two subspaces of ${\mathbb{R}}^n$ of respective dimensions $d$ and $e$ with $d+e⩽n$. The proximity between $A$ and $B$ is measured by $t=\min (d,e)$ canonical angles $0⩽{\theta }_1⩽\cdots ⩽{\theta }_t⩽\pi /2$; we set ${\psi }_j(A,B)=\sin {\theta }_j$. If $B$ is a rational subspace, his complexity is measured by its height $H(B)=\mathrm{covol}(B\cap {\mathbb{Z}}^n)$. We denote by ${\mu }_n(A|e{)}_j$ the exponent of approximation defined as the upper bound (possibly equal to $+\infty$) of the set of $\beta >0$ such that the inequality ${\psi }_j(A,B)⩽H(B{)}^{-\beta }$ holds for infinitely many rational subspaces $B$ of dimension $e$. We are interested in the minimal value ${\mathring{\mu }}_n(d|e{)}_j$ taken by ${\mu }_n(A|e{)}_j$ when $A$ ranges through the set of subspaces of dimension $d$ of ${\mathbb{R}}^n$ such that for all rational subspaces $B$ of dimension $e$ one has $\dim (A\cap B)<j$. We show that ${\mathring{\mu }}_4(2|2{)}_1=3$, ${\mathring{\mu }}_5(3|2{)}_1\le 6$ and ${\mathring{\mu }}_{2d}(d|\ell {)}_1⩽2d^2/(2d-\ell )$. We also prove a lower bound in the general case, which implies that ${\mathring{\mu }}_n(d|d{)}_d\underset{n\to +\infty }{\to }1/d$.

References in corpus (1)

• Approximation rationnelle de sous-espaces vectoriels