Defect of an octahedron in a rational lattice

Abstract

Consider an arbitrary $n$-dimensional lattice $\Lambda$ such that $\mathbb{Z}^n \subset \Lambda \subset \mathbb{Q}^n$. Such lattices are called {\it rational} and can always be obtained by adding $m \le n$ rational vectors to $\mathbb{Z}^n$. {\it Defect } $d({\cal E},\Lambda)$ of the standard basis $ {\cal E}$ of ${\mathbb Z}^n$ ($n$ 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 $L^1$-norm on $\mathbb{Q}^n$. Suppose that for each non-integer $x \in \Lambda$ inequality $||x|| > 1$ holds. Then the unit octahedron $O^n = \left\{{ x} \in \mathbb{R}^n: ||x|| \leqslant 1\right\}$ is called admissible with respect to $\Lambda$ and $d({\cal E},\Lambda)$ is also called defect of the octahedron $O^n$ with respect to $\cal{E}$ and is denoted as $d(O^n_{{\cal E}}, \Lambda)$. Let $ d_n^m = \max_{\Lambda \in {\cal A}_m} d(O^n_{{\cal E}},\Lambda), $ where $ {\cal A}_m $ is the set of all {\it rational} lattices that can be obtained by adding $m$ rational vectors to $\mathbb{Z}^n$: $ \Lambda = \left \langle {\mathbb Z}^n, { a}_1, \dots, { a}_m \right \rangle_{{\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 < 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.