Defect of an octahedron in a rational lattice

Abstract: Consider an arbitrary $n$-dimensional lattice $\Lambda$ such that $$\mathbb{Z}ⁿ \subset \Lambda \subset \mathbb{Q}<sup>n$$. Such lattices are called {\it rational} and can always be obtained by adding $m \le n$ rational vectors to $\mathbb{Z}<sup>n$. {\it Defect } $d({\cal E},\Lambda)$ of the standard basis ${\cal E}$ of ${\mathbb Z}<sup>n$ ($n$ unit vectors going in the directions of the coordinate axes) is defined as the smallest integer $d$ such that certain $(n-d)$ vectors from ${\cal E}$ together with some $d$ vectors from the lattice $\Lambda$ form a basis of $\Lambda$. Let $||...||$ be $L<sup>1$-norm on $\mathbb{Q}<sup>n$. Suppose that for each non-integer $x \in \Lambda$ inequality $||x|| &gt; 1$ holds. Then the unit octahedron $Oⁿ = \left{{ x} \in \mathbb{R}^n: ||x|| \leqslant 1\right}$ is called admissible with respect to $\Lambda$ and $d({\cal E},\Lambda)$ is also called defect of the octahedron $O<sup>n$ with respect to $\cal{E}$ and is denoted as $d(O^n_{{\cal E}}, \Lambda)$. Let $$d_n^m = \max_{\Lambda \in {\cal A}<em>m} d(Oⁿ</em>{{\cal E}},\Lambda),$$ where ${\cal A}<em>m$ is the set of all {\it rational} lattices that can be obtained by adding $m$ rational vectors to $\mathbb{Z}<sup>n$: $$\Lambda = \left \langle {\mathbb Z}^n, { a}_1, \dots, { a}_m \right \rangle</em>{{\mathbb Z}}, { a}_1, \dots, { a}_m \in {\mathbb Q}^n.$$ In this article we show that there exists an absolute positive constant $C$ such that for any $m &lt; n $ $$d_n^m \leq C \frac{n \ln (m+1)}{\ln \frac{n}{m}} \left(\ln\ln \left(\frac{n}{m}\right)^m \right)²$$ This bound was also claimed in $[1],[2]$, however the proof was incorrect. In this article along with giving correct proof we highlight substantial inaccuracies in those articles.