Arithmetic Difference of Middle Cantor Sets: Self-Similarity and Measure

Abstract

Let C be the space of all middle Cantor sets, it is shown that for a dense subset L of C × C × ℝ, set C α − λC β forms a uniformly contracting self-similar set when (C α, C β, λ) ∈ L. By selecting an appropriate element (C α C β, λ) ∈ L, we see that C α − λC β does not satisfy open set condition and then we calculate its Lebesgue measure that is zero.