Packing sets in Euclidean space by affine transformations

Abstract

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

27 pages