Upper bounds and spectrum for approximation exponents for subspaces of $\mathbb{R}^n$

Elio Joseph

Published 2021-06-11Version 1

This paper uses W. M. Schmidt's idea formulated in 1967 to generalise the classical theory of Diophantine approximation to subspaces of $\mathbb{R}^n$. Given two subspaces of $\mathbb{R}^n$ $A$ and $B$ 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 for infinitely many rational subspaces $B$ of dimension $e$, the inequality $\psi_j(A,B)\leqslant H(B)^{-\beta}$ holds. 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 if $A$ is included in a rational subspace $F$ of dimension $k$, its exponent in $\mathbb{R}^n$ is the same as its exponent in $\mathbb{R}^k$ via a rational isomorphism $F\to\mathbb{R}^k$. This allows us to deduce new upper bounds for $\mathring{\mu}_n(d\vert e)_j$. We also study the values taken by $\mu_n(A\vert e)_e$ when $A$ is a subspace of $\mathbb{R}^n$ satisfying $\dim(A\cap B)<e$ for all rational subspaces $B$ of dimension $e$.