Let $\Lambda \subset\mathbb{R}^n$ be a lattice satisfying $\|x-y\|\_2^2 \in \mathbb{Z}$ for all $x,y\in\Lambda$. How small can $\text{vol}(\Lambda)=\det(\Lambda)$ be?


For example, in dimension $2$, the hexagonal lattice has smallest volume of any integral lattice, equal to $\frac{\sqrt{3}}{2}$. In higher dimensions, rescaling an even unimodular lattice by a factor of $\sqrt{2}$ yields a lattice with integral distances with $\text{vol}(\Lambda) = 2^{-\frac{n}{2}}$.


What is the smallest volume possible as $n\rightarrow\infty$?



The smallest volume possible for a lattice with integer distances?
After scaling your lattice by $\sqrt{2}$, the Gram matrix has integral entries, so the absolute value of its determinant, being a nonzero integer, is at least $1$. This gives a lower bound of $2^{-n/2}$.


