  The aim of this article is to extend previous works about the asymptotics of
an ill-prepared fast rotating, highly stratified incompressible Navier-Stokes
system. Thanks to improved Strichartz estimates, we are able not only to cover
a case which was unanswered in our previous work (allowing bigger ill-prepared
initial data) but also to reduce some assumptions on the initial data. In
passing we also widen the range of some parameters and compare two methods to
obtain dispersive estimates.
