Before revision:  We study two cases of covering sets: c-approximate r-nets and randomly shifted grids, and we discuss the tradeoff between them in terms of preprocessing time and target dimension  .
After revision:  We study two cases of covering sets: c-approximate r-nets and randomly shifted grids, and we discuss the tradeoff between them in terms of preprocessing time and target dimension . We employ Cauchy variables: certain concentration bounds derived should be of independent interest .