How likely can a point be in different Cantor sets

Given m is an element of N->= 2 let K = {K-lambda : lambda is an element of (0,1/m]} be a class of self-similar sets with each K-lambda = {Sigma(infinity)(i=1) d(i)lambda(i) : d(i) is an element of {0,1, ..., m - 1}, i >= 1). In this paper we investigate the likelyhood of a point in the self-similar sets of K. More precisely, for a given point x is an element of (0, 1) we consider the parameter set Lambda(x) = {lambda is an element of (0, 1/m] : x is an element of K-lambda}, and show that Lambda(x) is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets of Lambda(x) with large thickness we show that for any x, y is an element of (0, 1) the intersection Lambda(x) boolean AND Lambda(y) also has full Hausdorff dimension.