### A ratio ergodic theorem for multiparameter non-singular actions

We prove a ratio ergodic theorem for non-singular free ${\mathbb{Z}}^{d}$ and ${\mathbb{R}}^{d}$ actions, along balls in an arbitrary norm. Using a Chacon–Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in ${\mathbb{R}}^{d}$. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact that boundaries of balls have lower dimension than the ambient space. We also show that for general group...