Minkowski’s Lattice Point Theorem

We present a proof of a famous theorem of Hermann Minkowski which spawned a beautiful branch of number theory known as the Geometry of Numbers. This theorem gives a condition on the volume of a centrally symmetric convex body in n-dimensional Euclidean space to contain at least one lattice point except the trivial point \Vec{0}.

Definitions and Notations

Definition 1 In an n-dimensional vector space V over the field \mathbb{R}, a lattice is a free abelian subgroup of V of the following form


where v_1,v_2,\ldots,v_m are linearly independent vectors in V. Then \Gamma is a free abelian group of rank m. A lattice is said to be complete if m=n. Assuming \Gamma to be complete, we define the fundamental mesh in \Gamma with respect the the v_i to be

\displaystyle \Phi=\{x_1v_1+x_2v_2+\cdots+x_nv_n:x_i\in\mathbb{R},x_i\in[0,1),1\leq i\leq n\}

The volume of the lattice \Gamma is defined to be the volume of the fundamental mesh \Phi denoted as \mathrm{Vol}(\Phi), i.e. \mathrm{Vol}(\Gamma)=\mathrm{Vol}(\Phi). There are a few equivalent expressions for this volume. One expression convenient for computations is given by the following relation

\mathrm{Vol}(\Phi)=\sqrt{\det[\langle v_i,v_j\rangle]}

where \langle \cdot,\cdot\rangle is the usual inner-product (or a symmetric positive definite bilinear form) defined on the vector space V.

Definition 2 (Convex and Centrally Symmetric Body)

A subset X of V is said to be centrally symmetric if for any point x\in X we have -x\in X, and the subset is convex if for any two distinct points x,y in X the line segment \{tx+(1-t)y:t\in[0,1]\} is contained in X. For example, a ball is convex in \mathbb{R}^3 but a solid torus is not.

Minkowski’s Lattice Point Theorem

Theorem: Let V be an n-dimensional Euclidean vector space (\mathbb{R}-vector space) and \Gamma be a complete lattice of V. Let X be a centrally symmetric convex subset of V such that


Then X contains at least one point other than \vec{0} in \Gamma.

Proof: Consider the dilation \frac{1}{2}X=\left\{\frac{1}{2}x:x\in X\right\}. For any \gamma\in \Gamma, consider the translated sets \frac{1}{2}X+\gamma. We will show that there exist \gamma_1,\gamma_2 \in \Gamma such that \gamma_1\neq\gamma_2 and


For the sake of contradiction, assume the sets \frac{1}{2}X+\gamma,\gamma\in\Gamma are pairwise disjoint. Then the intersections \Phi\cap\left(\frac{1}{2}X+\gamma\right),\gamma\in\Gamma are also disjoint for a fundamental mesh \Phi. This gives us the following inequality involving volumes

\mathrm{Vol}(\Phi)\geq\sum_{\gamma\in \Gamma}\mathrm{Vol}\left(\Phi\cap(\frac{1}{2}X+\gamma)\right)

We know that translation preserves volumes, so we consider the following translations


Hence for each \gamma\in \Gamma, we have


We claim that the \Phi-\gamma cover V as \gamma varies over \Gamma. Let x\in V. Since \{v_1,v_2,\ldots,v_n\} is a set of n linearly independent vectors in V and \mathrm{dim}(V)=n, it follows that \{v_1,v_2,\ldots,v_n\} is a basis of V. Then \exists \lambda_1,\lambda_2,\ldots,\lambda_n\in\mathbb{R} such that x=\sum_{i=1}^{n}\lambda_iv_i. We know that every real number r can be written as r=\lfloor r\rfloor+\{r\} where \lfloor r\rfloor\in\mathbb{Z} and \{r\}\in[0,1). Then

x=\sum_{i=1}^{n}\lfloor\lambda_i\rfloor v_i+\sum_{i=1}^{n}\{\lambda_i\}v_i

Note that \sum_{i=1}^{n}\lfloor\lambda_i\rfloor v_i\in\Gamma and \sum_{i=1}^{n}{\lambda_i}v_i\in\Phi. Taking \gamma'=-\sum_{i=1}^{n}\lfloor\lambda_i\rfloor v_i, we observe that x\in\Phi-\gamma'. Hence, V\subset\bigcup_{\gamma\in\Gamma}(\Phi-\gamma). Since (\Phi-\gamma)\subset V for all \gamma\in\Gamma, we have \bigcup_{\gamma\in\Gamma}(\Phi-\gamma)\subset V. Hence \bigcup_{\gamma\in\Gamma}(\Phi-\gamma)=V, and so the sets (\Phi-\gamma)\cap\frac{1}{2}X cover \frac{1}{2}X. Therefore, we finally have


which is a contradiction to our initial assumption since \mathrm{Vol}(\Gamma)=\mathrm{Vol}(\Phi). Hence, we can choose \gamma_1,\gamma_2\in\Gamma, \gamma_1\neq\gamma_2 such that


Therefore, there exist x_1,x_2\in X, x_1 \neq x_2, such that


Since X is centrally symmetric and convex, -x_2\in X and thus \gamma_0=\gamma_1-\gamma_2=\frac{1}{2}x_1-\frac{1}{2}x_2\in X Therefore \gamma_0\neq\Vec{0} and \gamma_0\in\Gamma\cap X. This completes the proof of the theorem.


Since \gamma_0\neq\Vec{0} we also have -\gamma_0\in\Gamma\cap X and -\gamma_0\neq\gamma_0. This means X contains at least two distinct lattice points. What else can you find hidden in the proof?

A Useful Corollary

Taking V=\mathbb{R}^n and \Gamma=\mathbb{Z}^n, we observe that any convex centrally symmetric body in \mathbb{R}^n of volume strictly bigger that 2^n contains at least one point with integer coordinates other than \Vec{0}.

Minkowski’s Theorem in Action

Now we present a number theoretic application of Minkowski’s theorem. The following result was proved by Axel Thue using the pigeonhole principle. We give a proof using Minkowski’s lattice point theorem.

Theorem: Primes of the form 4k+1 can be expressed as a sum of two squares.

Proof. Let p be a prime of the form 4k+1. Then -1 is a quadratic residue modulo p or, equivalently, there exists a\in\mathbb{Z} such that a^2+1\equiv0\pmod{p}. Consider the two vectors v_1=(p,0) and v_2=(a,1) in \mathbb{R}^2. Let \alpha v_1+\beta v_2=(0,0) for some \alpha,\beta\in\mathbb{R}. This gives us p\alpha+a\beta=0,\beta=0 and hence \alpha=\beta=0. Therefore v_1,v_2 are linearly independent. Then \Gamma=v_1\mathbb{Z}+v_2\mathbb{Z} is a complete lattice in \mathbb{R}^2 with \mathrm{Vol}(\Gamma)=p.

Let (x,y)\in\Gamma. There exist A,B\in\mathbb{Z} such that (x,y)=Av_1+Bv_2. This implies x=Ap+Ba,y=B. Hence,

x^2+y^2=(Ap)^2+2ABpa+(Ba)^2+B^2\equiv B^2(a^2+1)\equiv0\pmod{p}

Consider the open disc D of radius \sqrt{2p} centered at the origin (0,0). We have, \mathrm{Vol}(D)=\mathrm{area}(D)=\pi(\sqrt{2p})^2=2\pi p>4p=2^2\mathrm{Vol}(\Gamma)

Thus, D is convex and centrally symmetric. By Minkowski’s theorem, there exists a lattice point apart from the origin in D. Let this point be (m,n). Then 0<m^2+n^2<(\sqrt{2p})^2=2p and p\mid (m^2+n^2) and hence m^2+n^2=p. We are done!


[1] Andreescu, T. and Dospinescu, G., 2008. Problems from the Book.

[2] Neukirch, J., 2013. Algebraic number theory (Vol. 322). Springer Science & Business Media.

[3] Cassels, J.W.S., 2012. An introduction to the geometry of numbers. Springer Science & Business Media.

Welcome to the Blog!

This blog is meant for students and faculty to learn or present interesting ideas in number theory and algebraic geometry. There will be two main types of posts, introductory learning posts, and expository research posts. The introductory learning posts will focus on developing a beginner’s understanding of a certain subject or area. Then the expository research posts are meant to expose people to modern mathematics and to allow readers to get a glimpse of new problems and ideas.

If you are a student or a professor who has done interesting reading, an REU, or a research project in number theory or algebraic geometry, we encourage you to post on this blog. If you want to post, please fill out the following form https://forms.gle/9eMdzDcWyEmM7TuJ8.

Create your website at WordPress.com
Get started