  We construct a union of N parallelograms of dimensions approximately 1/N x 1
in the plane, with the slope of their long sides in the standard Cantor set.
The union has area 1/log N but the union of the doubles has area log log N/ log
N. In particular, this implies unbounded of the associated maximal operator in
L^p for any p different from infinity.
  The construction is by randomizing an earlier construction of the second
author for the L^2 case. The proof that the construction satisfies the desired
conditions is by elementary estimates in the theory of percolation on trees as
developed by R. Lyons.
