Packing sets in Euclidean space by affine transformations

TL;DR

This paper studies how the Lebesgue measure and Hausdorff dimension of sets in Euclidean space change under affine transformations, including rigid motions, dilations, and similarities, with sharp thresholds identified.

Contribution

It provides new results on the measure and dimension of transformed sets, extending previous work to more general affine transformations with sharp dimensional thresholds.

Findings

Derived sharp thresholds for measure and dimension under affine transformations.

Extended analysis to dilations, translations, and similarity transformations.

Generalized previous results from smooth objects to broader classes of sets.

Abstract

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.