Multiplication on self-similar sets with overlaps

Abstract 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 .