  We adapt techniques of Hochman to prove a non-singular ergodic theorem for
$\mathbb{Z}^d$-actions where the sums are over rectangles with side lengths
increasing at arbitrary rates, and in particular are not necessarily balls of a
norm. This result is applied to show that the critical dimensions with respect
to sequences of such rectangles are invariants of metric isomorphism. These
invariants are calculated for a class of product actions.
