Blichfeldt's theorem

Remarks

Let $P$ be a fundamental parallelepiped of $\Lambda$, so that the translates $P + v$ with $v \in \Lambda$ are pairwise disjoint and cover $\R^n$. Cut $S$ into the pieces $S \cap (P + v)$ and translate each piece by $-v$ into $P$. The translated pieces have total volume $\operatorname{vol}(S) > \operatorname{vol}(P) = \det \Lambda$, so two of them overlap; a common point gives $x - v = y - w$ with $x \ne y$ in $S$ and $v \ne w$ in $\Lambda$. Minkowski's theorem follows: if $K$ is convex, symmetric about $0$, and $\operatorname{vol}(K) > 2^n \det \Lambda$, apply the theorem to $S = \tfrac12 K$. It gives $x \ne y$ in $S$, and $x - y = \tfrac12(2x) + \tfrac12(-2y)$ is a nonzero point of $K \cap \Lambda$. The bound is sharp: the half-open square $[0, 1)^2$ has area $1 = \det \Z^2$, and no two of its distinct points differ by an integer vector. More generally, if $\operatorname{vol}(S) > k \det \Lambda$ for a positive integer $k$, the same argument gives $k + 1$ distinct points of $S$ whose pairwise differences lie in $\Lambda$.

Going deeper

For a student

Blichfeldt's theorem is the measure-theoretic pigeonhole principle behind the geometry of numbers. It gives a short proof of Minkowski's convex body theorem, and through it of results such as the two-square and four-square theorems. It holds for any measurable set, convex or not, which makes it useful when the region of interest is not convex.