Bounds on the defect of an octahedron in a rational lattice (Q6180569)

In what follows let \(\Gamma\subset \mathbb{R}^n\) be an an n-dimensional lattice in Euclidean space containing the origin, with basis \(\xi=(e_1,e_2,\ldots,e_n)\). If \(\Gamma\) is a sub-lattice of the lattice \(\Lambda\), then \(\Lambda\) is referred to as a centering of the lattice \(\Gamma\). Furthermore define the defect of the basis \(\xi\), with respect to the lattice \(\Lambda\), to be the smallest integer \(d\) such that \((n-d)\) vectors chosen from \(\xi\), together with \(d\) vectors chosen from the lattice \(\Lambda\) form a basis for \(\Lambda\). The defect of the basis is denoted by \(d(\xi, \Lambda)\) or simply \(d\). For a given basis \(\xi\), an octahedron corresponding to \(\xi\) is defined as the set \[ \mathcal{O}_\xi^n=\left \{ x\in \mathbb{R}^n: x=\lambda_1 e_1+\ldots +\lambda_n e_n; |\lambda_1|+\ldots +|\lambda_n|\leq 1\right \}. \] The octahedron \(\mathcal{O}_\xi^n\) is called admissible with respect to the lattice \(\Lambda\) if its interior contains no points of \(\Lambda\), except for the origin and \(\pm e_i,\,\,\, 1\leq i\leq n\). For an octahedron \(\mathcal{O}_\xi^n\), corresponding to the basis \(\xi\), that is admissible with respect to the centering lattic \(\Lambda\), the quantity \(d(\xi, \Lambda)\) is denoted by \(d(\mathcal{O}_\xi^n, \Lambda)\) and is referred to as the defect of the admissible octahedron \(\mathcal{O}_\xi^n\) in the lattice \(\Lambda\). In this present work the author considers the case \(\Gamma=\mathbb{Z}^n\), with \(\xi\) the standard unit basis in the directions of the coordinate axes. They are interested in the property \[ d_n = \text{max}_{\Lambda \in \mathcal{A}_n} d(\mathcal{O}_\xi^n, \Lambda), \] where \(\mathcal{A}_n\) is the set of all rational lattices \(\Lambda\) such that \(\mathcal{O}_\xi^n\) is admissible with respect to \(\Lambda\). In conjunction with the result of Konyagin's, which says that there exists a positive constant \(C\) such that \(n-d_n \leq C \log n\), the author shows that \[ n- d_n = O(\log n). \] Further bounds on generalisations of \(d_n\) are also established.