For a very simple family of self-similar sets with two pieces, we prove, using a technique of Solomyak, that the intersection of the pieces can be a Cantor set with any dimension in
[
0
,
0.2
]
[0,0.2]
as well as a finite set of any cardinality
2
m
2^m
. The main point is that the open set condition is fulfilled for all these examples.