Defect of an octahedron in a rational lattice

Abstract

Consider an arbitrary n-dimensional lattice Λ such that Zn⊂Λ⊂Qn. Such lattices are called rational and can always be obtained by adding m⩽n rational vectors to Zn. The defectd(E,Λ) of the standard basis E of Zn (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 E together with some d vectors from the lattice Λ form a basis of Λ.Let ||⋅|| be L1-norm on Qn. Suppose that for each non-integer x∈Λ inequality ||x||>1 holds. Then the unit octahedron On=x∈Rn:||x||⩽1 is called admissible with respect to Λ and d(E,Λ) is also called the defect of the octahedron On with respect to E and is denoted by d(OEn,Λ).Let dnm=maxΛ∈Amd(OEn,Λ), where Am is the set of all rational lattices Λ such that (1) On is admissible w.r.t. Λ and (2) Λ can be obtained by adding m rational vectors to Zn: Λ=Zn,a1,…,amZ for some a1,…,am∈Qn. In this article we show that there exists an absolute positive constant C such that for any m<ndnm⩽Cnln(m+1)lnnmlnlnnmm2This bound was also claimed in [2] and [1], however the proof was incorrect. In this article along with giving correct proof we highlight substantial inaccuracies in those articles.