Theorem

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