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

TL;DR

This paper extends Diophantine approximation theory to subspaces of b^n, establishing bounds on approximation exponents based on subspace angles, rationality, and intersections, with implications for understanding subspace approximation complexity.

Contribution

It generalizes classical Diophantine approximation to subspaces, providing new bounds and invariance results for approximation exponents involving rational subspaces.

Findings

Derived upper bounds for minimal approximation exponents.

Proved invariance of exponents under rational isomorphisms.

Analyzed exponents for subspaces with specific intersection properties.

Abstract

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 $e$.