  We study the Degrees-of-Freedom (DoF) in a wireless setting in which K
Transmitters (TXs) aim at jointly serving K users. The performance is studied
when the TXs are faced with a distributed Channel State Information (CSI)
configuration in which each TX has access to its own multi-user imperfect
channel estimate based on which it designs its transmit coefficients. The
channel estimates are not only imperfectly acquired but they are also
imperfectly shared between the TXs. Our first contribution consists of
computing a genie-aided upper bound for the DoF of that setting. Our main
contribution is then to develop a new robust transmission scheme that leverages
the different qualities of CSI available at the TXs to improve the achieved
DoF. We show the surprising result that there is a CSI regime, coined the
Weak-CSIT regime, in which the genie-aided upper bound is achieved by the
proposed transmission scheme. Interestingly, the optimal DoF in the Weak-CSIT
regime only depends on the CSI quality at the best informed TX and not on the
CSI quality at all other TXs.
