Random sequences of integers, Sidon sets, density in the Bohr group, and sets of analyticity

Random sequences of integers, Sidon sets, density in the Bohr group, and sets of analyticity. We study properties of a sequence A obtained by a random selection of integers n, where n epsilon Lambda with probability pi(n) independently of the other choices. We distinguish two cases: if lim sup(n ->infinity)n pi(n) < infinity, A is a.s. a Sidon set, non-dense in the Bohr group; if lim(n ->infinity) n pi(n) = infinity, then Lambda is a.s. a set of analyticity and is dense in the Bohr group.