Universal Counting of Lattice Points in Polytopes

Given a lattice polytope $P$ (with underlying lattice $\text{lo}$), the universal counting function ${\text{uu}}_P({\text{lo}}^{\mathrm{\prime }})=\mathrm{\mid }P\cap {\text{lo}}^{\mathrm{\prime }}\mathrm{\mid }$ is defined on all lattices ${\text{lo}}^{\mathrm{\prime }}$ containing $\text{lo}$. Motivated by questions concerning lattice polytopes and the Ehrhart polynomial, we study the equation ${\text{uu}}_P={\text{uu}}_Q$.