  Can we find a self-similar set on the line with positive Lebesgue measure and
empty interior? Currently, we do not have the answer for this question for
deterministic self-similar sets. In this paper we answer this question
negatively for random self-similar sets which are defined with the construction
introduced in the paper Jordan, Pollicott and Simon (Commun. Math. Phys.,
2007). For the same type of random self-similar sets we prove the Palis-Takens
conjecture which asserts that at least typically the algebraic difference of
dynamically defined Cantor sets is either large in the sense that it contains
an interval or small in the sense that it is a set of zero Lebesgue measure.
