MULTIPLE REPRESENTATIONS OF REAL NUMBERS ON SELF-SIMILAR SETS WITH OVERLAPS

Abstract

Let [Formula: see text] be the attractor of the following iterated function system (IFS) [Formula: see text] where [Formula: see text] and [Formula: see text] is the convex hull of [Formula: see text]. The main results of this paper are as follows: [Formula: see text] if and only if [Formula: see text] where [Formula: see text]. If [Formula: see text], then [Formula: see text]As a consequence, we prove that the following conditions are equivalent:(1) For any [Formula: see text], there are some [Formula: see text] such that [Formula: see text].(2) For any [Formula: see text], there are some [Formula: see text] such that [Formula: see text](3) [Formula: see text].