Packing sets in Euclidean space by affine transformations

For Borel subsets Θ⊂ O(d)×ℝ^d (the set of all rigid motions) and E⊂ℝ^d, we define Θ(E):=⋃_(g,z)∈Θ(gE+z). In this paper, we investigate the Lebesgue measure and Hausdorff dimension of Θ(E) given the dimensions of the Borel sets E and Θ, when Θ has product form. We also study this question by replacing rigid motions with the class of dilations and translations; and similarity transformations. The dimensional thresholds are sharp. Our results are variants of some previously known results in the literature when E is restricted to smooth objects such as spheres, k-planes, and surfaces.