Multiplication on self-similar sets with overlaps.

Let A , B ⊂ R. Define A ⋅ B = { x ⋅ y : x ∈ A , y ∈ B }. In this paper, we consider the following class of self-similar sets with overlaps. Let K be the attractor of the IFS { f 1 (x) = λ x , f 2 (x) = λ x + c − λ , f 3 (x) = λ x + 1 − λ } , where f 1 (I) ∩ f 2 (I) ≠ ∅ , (f 1 (I) ∪ f 2 (I)) ∩ f 3 (I) = ∅ , and I = [ 0 , 1 ] is the convex hull of K. The main result of this paper is K ⋅ K = [ 0 , 1 ] if and only if (1 − λ) 2 ≤ c. Equivalently, we give a necessary and sufficient condition such that for any u ∈ [ 0 , 1 ] , there exist some x , y ∈ K such that u = x ⋅ y. [ABSTRACT FROM AUTHOR]