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!

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    $\begingroup$ 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
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    $\begingroup$ 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

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.

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