In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson.
"The algebraic difference of two random Cantor sets: The Larsson family." Ann. Probab. 39 (2) 549 - 586, March 2011. https://doi.org/10.1214/10-AOP558