On the packing dimension of unions and extensions of $k$-planes

We study the packing dimension of unions of subsets of $k$-planes in $\mathbb{R}^n$ using tools from algorithmic information theory, obtaining an analog of a result of Héra and a mild generalization of a recent result of Fraser. Along the way, we introduce a notion of effective dimension on the Grassmannian and affine Grassmannian, and we establish several useful algorithmic and geometric tools in this setting. Additionally, we consider how the packing dimension of the union of certain subsets of $k$-planes changes when the subsets are extended to the entire $k$-plane. Finally, we improve the above bounds for unions and extensions in the special case that $k=n-1$.