Approximating the SVP to within a factor (1-1/dimϵ) is NP-hard under randomized conditions

Recently M. Ajtai showed that to approximate the shortest lattice vector in the l₂-norm within a factor (1+2(-dim^k)), for a sufficiently large constant k, is NP-hard under randomized reductions. We improve this result to show that to approximate a shortest lattice vector within a factor (1+dim^-ε), for any ε>0, is NP-hard under randomized reductions. Our proof also works for arbitrary l_p-norms, 1⩽p<∞