Low-dimensional lattices V. Integral coordinates for integral lattices

We say that an n-dimensional (classically) integral lattice $\Lambda$ is s-integrable, for an integer s, if it can be described by vectors $s^{- \frac{1}{2}}(x_1, ..., x_k)$, with all $x_i \in \mathbb{Z}$, in a euclidean space of dimension $k \geqslant n$. Equivalently, $\Lambda$ is s-integrable if and only if any quadratic form f(x) corresponding to $\Lambda$ can be written as s$^{-1}$ times a sum of k squares of linear forms with integral coefficients, or again, if and only if the dual lattice $\Lambda^*$ contains a eutactic star of scale s. This paper gives many techniques for s-integrating low-dimensional lattices (such as E$_8$ and the Leech lattice). A particular result is that any one-dimensional lattice can be 1-integrated with k = 4: this is Lagrange's four-squares theorem. Let $\phi$(s) be the smallest dimension n in which there is an integral lattice that is not s-integrable. In 1937 Ko and Mordell showed that $\phi$(1) = 6. We prove that $\phi$(2) = 12, $\phi$(3) = 14, 21 $\leqslant$ $\phi$ (4) $\leqslant$ 25, 16 $\leqslant \phi$(5) $\leqslant$ 22, $\phi$(s) 4$\leqslant$ 4s + 2 (s odd), $\phi$(s) $\leqslant$ 2$\pi$es(1 + o(1)) (s even) and $\phi$(s) $\geqslant$ 2 $\ln \ln$ s/$\ln \ln \ln$ s(1 + o(1)).