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Abstract
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This paper follows the generalisation of the classical theory of Diophantine approximation to subspaces
of established by W.
M. Schmidt in 1967. Let
and
be two subspaces
of
of respective
dimensions
and
with
. The proximity
between
and
is measured
by
canonical
angles
;
we set
. If
is a rational subspace, its complexity is measured by its height
. We
denote by
the exponent of approximation defined as the upper bound (possibly equal to
) of the set of
such that the inequality
holds for infinitely many
rational subspaces
of
dimension
. We are interested
in the minimal value
taken by
when
ranges through the set of
subspaces of dimension
of
such that for all
rational subspaces
of dimension
one has
. We
show that
,
and
.
We also prove a lower bound in the general case, which implies that
as
.
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Keywords
Diophantine approximation, rational subspaces, number
theory, approximation exponents
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Mathematical Subject Classification
Primary: 11J13
Secondary: 11J25
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Milestones
Received: 8 June 2021
Revised: 31 December 2021
Accepted: 20 January 2022
Published: 30 March 2022
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