On covering systems over finite nilpotent groups and lattices

We prove a quantitative non-covering theorem for finite nilpotent groups. As a consequence, every finite covering of $\Z^d$ by affine full-rank sublattices with pairwise distinct indices contains a member of bounded index, uniformly in $d$. We also establish a summability criterion allowing repeated indices, which extends to finite groups containing a normal nilpotent subgroup of bounded index.