Packing sets in Euclidean space by affine transformations

For Borel subsets \(\Theta\subset O(d)\times \mathbb{R}^d\) (the set of all rigid motions) and \(E\subset \mathbb{R}^d\), we define \begin{align*} \Theta(E):=\bigcup_{(g,z)\in \Theta}(gE+z). \end{align*} In this paper, we investigate the Lebesgue measure and Hausdorff dimension of \(\Theta(E)\) given the dimensions of the Borel sets \(E\) and \(\Theta\), when \(\Theta\) 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.