Periodic structure of translational multi-tilings in the plane

Suppose $f\in L^1(\Bbb{R}^d)$, $\Lambda\subset\Bbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with weight. We prove there exists lattice $L\subset\Bbb{R}^d$ $f+L$ also tiles, possibly different As corollary, together result Kolountzakis, it implies any convex polygon multi-tiles the plane by translations admits multi-tiling, multiplicity.