2
$\begingroup$

Question: Suppose $\{v_1,\ldots,v_{n-1}\}$ is an integer basis for the lattice $$L=\{(x_1,\ldots,x_n)\in\mathbb{Z}^n: > a_1x_1+\cdots+a_nx_n=0\},$$ where the $a_i$ are fixed nonzero integers. Is the volume $V(P)=\det(L)$ (see this for a proof that they are equal) of its fundamental parallelotope $P=\{t_1v_1+\cdots+t_{n-1}v_{n-1} > \mid t_i\in[0,1)\}$ necessarily equal to $$\frac{\sqrt{a_1^2+\cdots+a_n^2}}{\gcd(a_1,\ldots,a_n)}?$$

I used the case $n=3$ along with Minkowski's theorem (in the geometry of numbers) to solve the following Miklos problem from 2000:

Let $a<b<c$ be positive integers. Prove that there exist integers $x,y,z$, not all zero, such that $ax+by+cz=0$ and $\max(|x|,|y|,|z|)\le > 1+\frac{2}{\sqrt3}\sqrt{c}$, and show that the constant $\frac{2}{\sqrt3}$ cannot be improved.

However, I was only able to find a brute force proof for this special case (see lemma 1 in my AoPS post here), and I'm not sure if it's as easy for larger values of $n$.

But I'm pretty sure this should be true in general (I've tried several cases for $n=4$ and $n=5$), so I would appreciate it if someone could give a (clean?) proof, reference, or counterexample. Thanks!

$\endgroup$
  • 4
    $\begingroup$ Without loss of generality the gcd is $1$. Then ${\bf Z}^n$ contains the direct sum $L_1$ of $L$ with ${\bf Z}\cdot(a_1,\ldots,a_n)$ with index $A := a_1^2+\cdots+a_n^2$. Thus a fundamental parallelotope for $L_1$ has volume $A$. But this volume is the product of the corresponding volumes for $L$ and ${\bf Z}\cdot(a_1,\ldots,a_n)$. The latter volume is $A^{1/2}$; hence the former is $A / A^{1/2} = A^{1/2}$, QED. $\endgroup$ – Noam D. Elkies Jul 26 '12 at 3:53
  • $\begingroup$ Again, wlog gcd is 1. Take a fundamental parallelotope of $L$ and add a vector $v_n$ to obtain a f.p. of ${\mathbb Z}^n$. Then the endpoints of $v_n$ lie in the ``neighboring'' affine hyperspaces parallel to $L$, and the distance between them is exactly $1/\sqrt{a_1^2+\dots+a_n^2}$. $\endgroup$ – Ilya Bogdanov Jul 26 '12 at 4:48
  • 5
    $\begingroup$ More generally: suppose $L_0$ is a lattice primitively embedded in ${\bf Z}^n$; that is, if $L_0$ contains $mv$ for some nonzero $m\in\bf Z$ and $v \in {\bf Z}^n$ then $L_0$ contains $v$. Let $L \subset {\bf Z}^n$ consist of the integer vectors orthogonal to every vector in $L_0$. Then the fundamental parallelotopes of the lattices $L_0$, $L$ have the same volume. Victor's question is the special case when $L_0$ is the rank-$1$ lattice ${\bf Z}\cdot(a_1,\ldots,a_n)$. $\endgroup$ – Noam D. Elkies Jul 26 '12 at 4:53
1
$\begingroup$

I borrowed a few books; it turns out the material provided in comments to the answer is not in print in many places. One is a book I have, Lattices and Codes by Wolfgang Ebeling, on page 5 in the second edition. In turn, he refers to Looijenga and Peters (1981), who devote section 2 of their article to lattices, the particular result is on page 154. I lost that link, hold on.

enter image description here enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.