Multiplication on self-similar sets with overlaps

Let A, B subset of R. Define A . B= {x.y:x is an element of A,y is an element of 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) = lambda x, f(2)(x) = lambda x + c - lambda, f(3)(x) = lambda x + 1 - lambda}, where f(1)(I) boolean AND f(2)(r) not equal 0, (f(1)(I)boolean OR f(2)(I)) boolean AND f(3)(I) = 0, 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 -lambda)(2) <= c. Equivalently, we give a necessary and sufficient condition such that for any u is an element of [0, 1], there exist some x, y E K such that u = x. y. (C) 2019 Elsevier Inc. All rights reserved.