Blichfeldt vs Minkowski Theorems (Mathematics of Lattices)

Question

I am trying to study and understand the basic theorems about mathematics of lattices. In particular, I understood statements and proofs of both Blichfeldt and Minkowski. My doubt was about their utility,

I mean: the first one (Blichfeldt) states that, given a set $S \subseteq \text{span}(S)$, we can find two points $\textbf{z}_1$ and $\textbf{z}_2$ such that $\textbf{z}_2 - \textbf{z}_1\in \Lambda$ (where $\Lambda$ is the lattice). So we bounded the length of the shortest nonzero vector into a set that can have a volume of at least $\text{det}(\Lambda)$.

Minkowski, on the other hand, bounds the length of the shortest nonzero vector into a set of volume at least $2^n\text{det}(\Lambda)$.

So, isn't Blichfeldt a stronger result, since the volume is smaller?

Answer 1

First let us look at the exact statements of the 2 theorems:

Blichfeld’s Theorem: For any full-rank lattice $\Lambda \subseteq \mathbb{R}^n$ and measurable set $S \subseteq \mathbb{R}^n$ with $\mathrm{vol}(S) > \mathrm{det}(\Lambda)$ there exist two distinct points $z_1, z_2 \in S$ such that $z_1 - z_2 \in \Lambda$.

However it does not give any guarantee about whether $z_1$ and $z_2$ will itself be lattice vectors, or if $z_1 - z_2$ is within $S$. In fact we can only bound $\|z_1 - z_2\| \le \|z_1\| + \|z_2\|$, where the equality holds when $z_2 = cz_1$ for some $c < 0$. Thus $z_1 - z_2$ might very well not belong to $S$ and hence contrary to your belief, Blichfeld does not really “bound the length of the shortest nonzero vector into a set that can have a volume of at least $\mathrm{det}(\Lambda)$”.

Minkowski’s Convex Body Theorem: For a full-rank lattice $\Lambda \subseteq \mathbb{R}^n$, and a centrally-symmetric convex set $S$, if $\mathrm{vol}(S)>2^n\mathrm{det}(\Lambda)$ then $S$ contains a nonzero lattice point.

Of course this theorem guarantees that the set $S$ itself contains a lattice vector, and hence leads to a concrete bound of the shortest vector. Note that to prove this theorem on might consider the set $\hat{S} = \{x | 2x \in S\}$, which satisfies the condition of Blichfeld’s theorem, and hence there exists $z_1, z_2 \in \hat{S}$ such that $z_1 - z_2 \in \Lambda$. Now from central-symmetry and convexity we get that $S$ itself contains $z_1 - z_2$, which is the lattice vector promised by Blichfeld. Thus Minkowski’s theorem does indeed follow from Blichfeld’s theorem, however when it comes to finding the shortest lattice vector or bounding its length, Minkowski says something concrete, while Blichfeld doesn’t.