Differences of random Cantor sets and lower spectral radii

We investigate the question under which conditions the algebraic difference between two independent random Cantor sets C-1 and C-2 almost surely contains an interval, and when not. The natural condition is whether the sum d(1) + d(2) of the Hausdorff dimensions of the sets is smaller (no interval) or larger (an interval) than 1. Palis conjectured that generically it should be true that d(1) + d(2) > 1 should imply that C-1 - C-2 contains an interval. We prove that for 2-adic random Cantor sets generated by a vector of probabilities (p(0), p(1)) the interior of the region where the Palis conjecture does not hold is given by those p(0), p(1) which satisfy p(0) + p(1) > root 2 and p(0)p(1)(1 + p(0)(2) + p(1)(2)) < 1. We furthermore prove a general result which characterizes the interval/no interval property in terms of the lower spectral radius of a set of 2 x 2 matrices.