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A conjecture made by Lapidoth, Shamai and Wigger at Allerton 2005 (also anopen problem presented at ITA 2006) states that the DoF of a 2 user broadcastchannel, where the transmitter is equipped with 2 antennas and each user isequipped with 1 antenna, must collapse under finite precision CSIT. In thiswork we prove that the conjecture is true in all non-degenerate settings (e.g.,where the probability density function of unknown channel coefficients existsand is bounded). The DoF collapse even when perfect channel knowledge for oneuser is available to the transmitter. This also settles a related recentconjecture by Tandon et al. The key to our proof is a bound on the number ofcodewords that can cast the same image (within noise distortion) at theundesired receiver whose channel is subject to finite precision CSIT, whileremaining resolvable at the desired receiver whose channel is precisely knownby the transmitter. We are also able to generalize the result along twodirections. First, if the peak of the probability density function is allowedto scale as O(P^(\alpha/2)), representing the concentration of probabilitydensity (improving CSIT) due to, e.g., quantized feedback at rate(\alpha/2)\log(P), then the DoF are bounded above by 1+\alpha, which is alsoachievable under quantized feedback. Second, we generalize the result to the Kuser broadcast channel with K antennas at the transmitter and a single antennaat each receiver. Here also the DoF collapse under non-degenerate channeluncertainty. The result directly implies a collapse of DoF to unity undernon-degenerate channel uncertainty for the general K-user interference and MxNuser X networks as well.

Aligned Image Sets under Channel Uncertainty: Settling a Conjecture by  Lapidoth, Shamai and Wigger on the Collapse of Degrees of Freedom under  Finite Precision CSIT