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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 realnumbers, typically as special values of analytic functions, how to prove thatthe $\mathbb{Q}$-vector space spanned by $1$ and those three numbers hasdimension at least 3, whenever we are unable to achieve full linearindependence, by using simultaneous approximations, i.e. those usually arisingfrom Hermite-Pad\'e approximations of type II and their suitablegeneralizations. It should be recalled that approximations of type I and II arerelated, at least in principle: when the numerical application consists inspecializing actual functional constructions of the two types, they can beobtained, 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 behaviorof the approximations at the infinite place of $\mathbb{Q}$. Ratherinterestingly, the two view-points split away regarding the asymptoticbehaviors at finite places (i.e. primes) of $\mathbb{Q}$, and this makes theuse of type II more convenient for particular purposes. In addition, sometimeswe know type II approximations to a given set of functions, for which type Iapproximations are not known explicitly. Our approach can be regarded as a dualversion of the standard linear independence criterion, which goes back toSiegel.

Vectors of type II Hermite-Pad\'e approximations and a new linear independence criterion