All types implies torsion

Abstract

We prove the following theorem. Given a positive integer n n and a subset A A of Z n {{\mathbf {Z}}^n} with the following properties: (1) for every ( a 1 , … , a n ) \left ( {{a_1}, \ldots ,{a_n}} \right ) in A A the inequality Σ i = 1 n | a i | ≥ 2 \Sigma _{i = 1}^n {\left | {{a_i}} \right |} \geq 2 holds, and (2) for every ( x 1 , … , x n ) \left ( {{x_1}, \ldots ,{x_n}} \right ) in R n {{\mathbf {R}}^n} there exists an ( a 1 , … , a n ) \left ( {{a_1}, \ldots ,{a_n}} \right ) in A A with a i x i ≥ 0 {a_i}{x_i} \geq 0 for i = 1 , … , n i = 1, \ldots ,n , there exists a subset A 0 {A_0} of A A such that Z n {{\mathbf {Z}}^n} modulo the subgroup generated by A 0 {A_0} contains a nontrivial torsion element.