Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Upper bounds and spectrum for approximation exponents for subspaces of
  $\mathbb{R}^n$   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 $e$.

1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: stat