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Rudolph showed that the orbits of any measurable, measure preserving $\mathbbR^d$ action can be measurably tiled by $2^d$ rectangles and asked if thisnumber of tiles is optimal for $d>1$. In this paper, using a tiling of $\mathbbR^d$ by notched cubes, we show that $d+1$ tiles suffice. Furthermore, using adetailed 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 completelypositive entropy this bound is optimal there exist mixing $\mathbb R^2$ actionswhose orbits can be tiled by 2 tiles.

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