  We propose a linear independence criterion, and outline an application of it.
Down to its simplest case, it aims at solving this problem: given three real
numbers, typically as special values of analytic functions, how to prove that
the $\mathbb{Q}$-vector space spanned by $1$ and those three numbers has
dimension at least 3, whenever we are unable to achieve full linear
independence, by using simultaneous approximations, i.e. those usually arising
from Hermite-Pad\'e approximations of type II and their suitable
generalizations. It should be recalled that approximations of type I and II are
related, at least in principle: when the numerical application consists in
specializing actual functional constructions of the two types, they can be
obtained, one from the other, as explained in a well-known paper by K.Mahler
[34]. That relation is reflected in a relation between the asymptotic behavior
of the approximations at the infinite place of $\mathbb{Q}$. Rather
interestingly, the two view-points split away regarding the asymptotic
behaviors at finite places (i.e. primes) of $\mathbb{Q}$, and this makes the
use of type II more convenient for particular purposes. In addition, sometimes
we know type II approximations to a given set of functions, for which type I
approximations are not known explicitly. Our approach can be regarded as a dual
version of the standard linear independence criterion, which goes back to
Siegel.
