  The Skorokhod Embedding problem is well understood when the underlying
process is a Brownian motion. We examine the problem when the underlying is the
simple symmetric random walk and when no external randomisation is allowed. We
prove that any measure on Z can be embedded by means of a minimal stopping
time. However, in sharp contrast to the Brownian setting, we show that the set
of measures which can be embedded in a uniformly integrable way is strictly
smaller then the set of centered probability measures: specifically it is a
fractal set which we characterise as an iterated function system. Finally, we
define the natural extension of several known constructions from the Brownian
setting and show that these constructions require us to further restrict the
sets of target laws.
