Abstract

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

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