The intersections of self-similar and self-affine sets with their perturbations under the weak separation condition

For a self-similar or self-affine iterated function system (IFS), let mu be the self-similar or self-affine measure and K be the self-similar or self-affine set. Assume that the IFS satisfies the weak separation condition and K is totally disconnected; then, by using the technique of neighborhood decomposition, we prove that there is a neighborhood Omega of the identity map Id such that sup {mu(g(K) boolean AND K): g is an element of Omega \ {Id}} < 1.