Simultaneously good bases of a lattice and its reciprocal lattice
Associate to an ordered basis \(B=[{\mathfrak b}_ 1,...,{\mathfrak b}_ n]\) of a lattice L the reciprocal basis \(B^*=[{\mathfrak b}^*_ 1,...,{\mathfrak b}^*_ n]\) of the reciprocal lattice \(L^*\) defined by \(<{\mathfrak b}_ i,{\mathfrak b}^*_{n+1-j}>=0\) if \(i\neq j\) and 1 otherwise. This paper studies the problem of choosing a basis B such that B and \(B^*\) simultaneously consist of short vectors. Let \(\lambda_ i(L)\), \(\lambda_ i(L^*)\) denote the successive minima of L and \(L^*\) respectively. It is shown that every lattice L has an ordered basis such that simultaneously \(\| {\mathfrak b}_ i\| \leq c(n)\lambda_ i(L)\) and \(\| {\mathfrak b}^*_ i\| \leq c(n)\lambda_ i(L^*)\) for \(1\leq i\leq n\), where \(c(n)=\exp (c_ 0n^{1/3})\) and \(c_ 0\) is a positive constant independent of n. This result is obtained by a study of the unipotent group N(n,\({\mathbb{R}})\) of all lower triangular matrices with diagonal entries equal to one, under the action of its discrete subgroup N(n,\({\mathbb{Z}})\). For any M in N(n,\({\mathbb{R}})\) there exists some W in N(n,\({\mathbb{Z}})\) such that \(Max(\| WM\|, \| (WM)^{-1}\|)\leq \exp (cn^{1/3}),\) where c is a positive constant independent of n.
scientific article; zbMATH DE number 4097485
scientific article