  Let F1 and F2 be independent copies of correlated fractal percolation, with
Hausdorff dimensions dimH(F1) and dimH(F2). Consider the following question:
does dimH(F1)+dimH(F2)>1 imply that their algebraic difference F1-F2 will
contain an interval? The well known Palis conjecture states that `generically'
this should be true. Recent work by Kuijvenhoven and the first author
(arXiv:0811.0525) on random Cantor sets can not answer this question as their
condition on the joint survival distributions of the generating process is not
satisfied by correlated fractal percolation. We develop a new condition which
permits us to solve the problem, and we prove that the condition of
(arXiv:0811.0525) implies our condition. Independently of this we give a
solution to the critical case, yielding that a strong version of the Palis
conjecture holds for fractal percolation and correlated fractal percolation:
the algebraic difference contains an interval almost surely if and only if the
sum of the Hausdorff dimensions of the random Cantor sets exceeds one.
