In this post, I wanted to give a proof of Minkowski’s Theorem on lattice points using a lemma of Blichfeldt. There’s an intuitive probabilistic characterization of the lemma, but very few sources seem to rigorously establish the full proof, a gap which I attempted to fill in below. I wrote this as a note for Prof. Barry Mazur’s algebraic number theory course in the spring of 2024; in a future post, I’ll indicate the standard consequence of Minkowski’s Theorem, which is finiteness of the class group of any algebraic number field.
. Let be a full-rank lattice in (a discrete, cocompact subgroup) with fundamental domain and a bounded, measurable set such that Then there exist such that .
We reduce to the case where , applying a linear transformation as needed. Haar measure on is just Lebesgue measure (which we denote here by ), and is translation-invariant. Then .
Write for the characteristic function of . This function is Lebesgue-integrable because is a measurable set. Let Because is bounded, it follows that is bounded, as there are finitely many nonzero terms for any given .
Now we integrate both sides of this expression over . Thanks to our previous remark, we may freely switch integral and sum because the summation is a finite sum of nonnegative terms, so
This implies that for some , which gives the desired two points.
. Let be a full-rank lattice with fundamental domain , and let be convex, centrally symmetric, and measurable. If , then contains some nonzero point of .
Note that so by Blichfeldt’s lemma, there exist distinct such that . and hence are both centrally symmetric, so . Because is convex, implies that , so thus we have found our desired nonzero element in .