Separation properties for self-conformal sets

For a one-to-one self-conformal contractive system {omegaj}(j = 1)(m) on R-d with attractor K and conformality dimension alpha, Peres et al. showed that the open set condition and strong open set condition are both equivalent to 0 < H-α (K) < infinity. We give a simple proof of this result as well as discuss some further properties related to the separation condition.