We adapt techniques developed by 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 the natural action of $\mathbb {Z}^d$ on a product of d measure spaces.

中文翻译：

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非奇异作用：矩形上的遍历定理，适用于临界尺寸

我们采用 Hochman 开发的技术来证明非奇异遍历定理$\mathbb {Z}^d$-总和超过边长以任意速率增加的矩形的动作，特别是不一定是标准球。该结果用于表明关于此类矩形序列的临界维度是度量同构的不变量。这些不变量是为自然作用计算的$\mathbb {Z}^d$在一个产品上d测量空间。