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.

Abstract A procedure for defining virtual spaces, and the periodic one-electron two-electron integrals, plane-wave second quantized Hamiltonians has been developed, it was validated using full configuration interaction (FCI) calculations, as well executions of variational quantum eigensolver (VQE) circuits on Quantinuum’s ion trap computers accessed through Microsoft’s Azure Quantum service. Th...