  We study the task of differentially private clustering. For several basic
clustering problems, including Euclidean DensestBall, 1-Cluster, k-means, and
k-median, we give efficient differentially private algorithms that achieve
essentially the same approximation ratios as those that can be obtained by any
non-private algorithm, while incurring only small additive errors. This
improves upon existing efficient algorithms that only achieve some large
constant approximation factors.
  Our results also imply an improved algorithm for the Sample and Aggregate
privacy framework. Furthermore, we show that one of the tools used in our
1-Cluster algorithm can be employed to get a faster quantum algorithm for
ClosestPair in a moderate number of dimensions.
