Single-exponential bounds for the smallest singular value of Vandermonde matrices in the sub-Rayleigh regime

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