Shrinking integer lattices (Q1896343)

Let \(\Lambda\) be a full sublattice of the integer lattice \(\mathbb{Z}^n\) in Euclidean \(n\)-space. The author then asks for the maximal \(r>0\) such that there is a sublattice \(\Lambda'\) of \(\mathbb{Z}^n\) isometric to \(r^{-1} \Lambda\). In 1990 \textit{J. Cremona} and \textit{S. Landau} [SIAM J. Discrete Math. 3, 338-348 (1990; Zbl 0711.51010)] gave an upper bound for \(r\) and showed that this bound is always attained if \(n\leq 4\). The author shows that this bound is also always attained for \(n=5\). He further gives counterexamples which show that the bound is not always attained for \(n=6\) or \(n\geq 8\).