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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] as well as a finite set of any cardinality 2 m . The main point is that the open set condition is fulfilled for all these examples.

Self-similar sets with an open set condition and great variety of overlaps