Bounds on the 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 OEn=x∈Rn:‖x‖⩽1 is called admissible with respect to Λ and d(E,Λ) is also called the defect of the octahedron OEn with respect to E and is denoted by d(OEn,Λ).Let dn=maxΛ∈And(OEn,Λ), where An is the set of all rational lattices Λ such that OEn is admissible w.r.t. Λ. In this article we show that n−dn=Θ(logn).Let dnm=maxΛ∈Amd(OEn,Λ), where Am is the set of all rational lattices Λ such that 1) OEn 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 for any 0<ϵ<1 and large enough n we have m<nϵ8(1ϵ+1)⟹dnm<n−n1−ϵ.Finally we show that for any B>0 there exists a positive constant C>0 such that m<Cnloglognlog2n⟹dnm<n−Blogn.