  This article proves that, in terms of local times, the rescaled and
recentered cover times of finite subsets of the discrete cylinder by simple
random walk converge in law to the Gumbel distribution, as the cardinality of
the set goes to infinity. As applications we obtain several other results
related to covering in the discrete cylinder. Our method is new and involves
random interlacements, which were introduced by Sznitman in arXiv:0704.2560. To
enable the proof we develop a new stronger coupling of simple random walk in
the cylinder and random interlacements, which is also of independent interest.
