Weighted geometric set multi-cover via quasi-uniform sampling

We give a randomized polynomial time algorithm with approximation ratio $O(\log \phi(n))$ for weighted set multi-cover instances with a shallow cell complexity of at most $f(z,k) =z\phi(z) k^{O(1)}$. Up to constant factors, this matches a recent result of Chan, Grant, Konemann and Sharpe for the set cover case, i.e. when all the covering requirements are 1. One consequence of this is an $O(1)$-approximation for geometric weighted set multi-cover problems when the geometric objects have linear union complexity; for example when the objects are disks, unit cubes or halfspaces in $\mathbb{R}^3$. Another consequence is to show that the real diculty of many natural capacitated set covering problems lies with solving the associated priority cover problem only, and not with the associated multi-cover problem