  Let A be a subset of the real line. We study the fractal dimensions of the
k-fold iterated sumsets kA, defined as kA = A+...+A (k times).
  We show that for any non-decreasing sequence {a_k} taking values in [0,1],
there exists a compact set A such that kA has Hausdorff dimension a_k for all
k. We also show how to control various kinds of dimension simultaneously for
families of iterated sumsets.
  These results are in stark contrast to the Plunnecke-Rusza inequalities in
additive combinatorics. However, for lower box-counting dimension, the analogue
of the Plunnecke-Rusza inequalities does hold.
