  Recently, mass transference principles in metric number theory extend towards
two direction. On one hand, the shape of the approximating sets can be taken of
various shape, balls, rectangles or even general open sets (one refers to some
results of Rams and Koivusalo regarding this last example) when the ambient
measure is Lebesgue, on the other hand, progress have been made to understand
what can be said when the ambient measure is changed. For instance some
computation in the case of a self-similar measure with open set condition have
been made by Barral and Seuret. This article mixes the two approaches and
presents a ubiquity theorem which handles the case where the approximating sets
are rectangles and the measure is quasi-Bernoulli, but fully supported.
