Exercise 4, § 1

Let $H$ and $K$ be two discrete subgroups of $\mathbf{R}^n$ such that $H + K$ is a closed subgroup. Show that $H + K$ is then discrete and that $r(H) + r(K) = r(H \cap K) + r(H + K)$. (Let $V$ be the vector subspace of $\mathbf{R}^n$ generated by $H \cap K$; split up $H$ into the direct sum of $V \cap H$ and a discrete group $H_1$, and $K$ into the direct sum of $V \cap K$ and a discrete group $K_1$, using Exercise 3; then show that the sum $H_1 + K_1$ is direct and use Exercise 2.)