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

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 $d+e\leqslant n$. The proximity between $A$ and $B$ is measured by $t=\min(d,e)$ canonical angles $0\leqslant \theta_1\leqslant \cdots\leqslant \theta_t\leqslant \pi/2$; 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 $\mu_n(A\vert e)_j$ the exponent of approximation defined as the upper bound (possibly equal to $+\infty$) of the set of $\beta>0$ such that the inequality $\psi_j(A,B)\leqslant H(B)^{-\beta}$ holds for infinitely many rational subspaces $B$ of dimension $e$. We are interested in the minimal value $\mathring{\mu}_n(d\vert e)_j$ taken by $\mu_n(A\vert e)_j$ 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 $\mathring{\mu}_4(2\vert 2)_1=3$, $\mathring{\mu}_5(3\vert 2)_1\le 6$ and $\mathring{\mu}_{2d}(d\vert \ell)_1\leqslant 2d^2/(2d-\ell)$. We also prove a lower bound in the general case, which implies that $\mathring{\mu}_n(d\vert d)_d\xrightarrow[n\to+\infty]{} 1/d$.