Rudolph's Two-Step Coding Theorem and Alpern's Lemma for R^d Actions

Rudolph showed that the orbits of any measurable, measure preserving $\mathbb R^d$actioncanbemeasurablytiledby$2^d$ rectangles and asked if this number of tiles is optimal for $d>1$. In this paper, using a tiling of $\mathbb R^d$bynotchedcubes,weshowthat$d+1$ tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of ${\mathbb{R}}^2$ by two rectangles, we show that while for ${\mathbb{R}}^2$ actions with completely positive entropy this bound is optimal there exist mixing ${\mathbb{R}}^2$ actions whose orbits can be tiled by 2 tiles.