The purpose of this work is to lift the notion of restricted orbit equivalence to the category of free and ergodic actions of discrete amenable groups. We mean lift in two senses. First of course we will generalize the results in, where Rudolph developed a theory of restricted orbit equivalence for ℤ-actions, and in where both authors later established a similar theory for actions of ℤd, d ≥ 1 to actions of these more general groups. However, we will also lift in the sense that we will develop the axiomatics and argument structures in what we feel is a far more natural and robust fashion. Both and were based on axiomatizations of a notion called a “size” measuring the degree of distortion of a box in ℤd caused by a permutation. It is not evident that on their common ground, ℤ-actions, these two theories agree. Hence we refer to the first as a 1-size and the second as a p-size, p for “permutation”.
Here we will establish the axiomatics of what we will simply call a size. We ask that the reader accept this new definition. In the Appendix we show that any equivalence relation that arose from a p-size will arise from a size as we define it here. The same is not done for 1-sizes, but for a slight strengthening of this axiomatics that includes all the examples in.