Sample complexity of the distinct elements problem

We consider the distinct elements problem, where the goal is to estimate the number of distinct colors in an urn containing $k$ balls from repeated draws. We propose an estimator, based on sampling without replacement, with additive error guarantee. The sample complexity is optimal within $O(\log\log k)$ factors, and in fact within constant factors for most accuracy parameters. The optimal sample complexity is also applicable to sampling without replacement provided the sample size is a vanishing fraction of the urn size. One of the key auxiliary results is a sharp bound on the minimum singular values of a real rectangular Vandermonde matrix, which might be of independent interest.