Rudolph’s two step coding theorem and alpern’s lemma for ℝ^dactions

Rudolph showed that the orbits of any measurable, measure preserving Rd action can be measurably tiled by 2^drectangles and asked if this number of tiles is optimal for d > 1. In this paper, using a tiling of ℝ^dby notched cubes, we show that d + 1 tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of ℝ²by two rectangles, we show that while for ℝ²actions with completely positive entropy this bound is optimal, there exist mixing ℝ²actions whose orbits can be tiled by 2 tiles.