  We study a class of radially symmetric Coulomb gas ensembles at inverse
temperature $\beta=2$, for which the droplet consists of a number of concentric
annuli, having at least one bounded ``gap'' $G$, i.e., a connected component of
the complement of the droplet, which disconnects the droplet. Let $n$ be the
total number of particles. Among other things, we deduce fine asymptotics as $n
\to \infty$ for the edge density and the correlation kernel near the gap, as
well as for the cumulant generating function of fluctuations of smooth linear
statistics. We typically find an oscillatory behaviour in the distribution of
particles which fall near the edge of the gap. These oscillations are given
explicitly in terms of a discrete Gaussian distribution, weighted Szeg\H{o}
kernels, and the Jacobi theta function, which depend on the parameter $n$.
