  Following recent interest by the community, the scaling of the minimal
singular value of a Vandermonde matrix with nodes forming clusters on the
length scale of Rayleigh distance on the complex unit circle is studied. Using
approximation theoretic properties of exponential sums, we show that the decay
is only single exponential in the size of the largest cluster, and the bound
holds for arbitrary small minimal separation distance. We also obtain a
generalization of well-known bounds on the smallest eigenvalue of the
generalized prolate matrix in the multi-cluster geometry. Finally, the results
are extended to the entire spectrum.
