  Let C be a Cantor set. For a real number t let C+t be the translate of C by
t, We say two real numbers s,t are equivalent if the intersection of C and C+s
is a translate of the intersection of C and C+t. We consider a class of Cantor
sets determined by similarities with one fixed positive contraction ratio. For
this class of Cantor set, we show that an "initial segment" of the intersection
of C and C+t is a self-similar set with contraction ratios that are powers of
the contraction ratio used to describe C as a self- similar set if and only if
t is equivalent to a rational number. Our results are new even for the middle
thirds Cantor set.
