  We investigate and quantify the distinction between rectifiable and purely
unrectifiable 1-sets in the plane. That is, given that purely unrectifiable
1-sets always have null intersections with Lipschitz images, we ask whether
these sets intersect with Lipschitz images at a dimension that is close to one.
In an answer to this question, we show that one-dimensional attractors of
iterated function systems that satisfy the open set condition have subsets of
dimension arbitrarily close to one that can be covered by Lipschitz graphs.
Moreover, the Lipschitz constant of such graphs depends explicitly on the
difference between the dimension of the original set and the subset that
intersects with the graph.
