Multiplication on self-similar sets with overlaps

Arithmetic on Moran Sets

“…For more results, see [14]. In [25], Tian et al defined a class of overlapping self-similar sets as follows: let K be the attractor of the IFS…”

Interiors of continuous images of the middle-third Cantor set

“…The following Lemma 1 comes from [2] and [19], here we give its proof just for the self-containedness of the paper.…”

“…where L denotes the Lebesgue measure and C • C = {xy : x, y ∈ C}. One can find more results in [7,15,19,13,12] for the arithmetic representations of real numbers. Motivated by Athreya, Reznick and Tyson's result, it is natural to ask whether C • C contains a non-empty interior.…”

Multiple Representations of Real Numbers on Self-Similar Sets With Overlaps

“…In [28], Jiang, Wang and Xi gave a necessary condition such that when c = λ − λ 2 the self-similar set K is bi-Lipschitz equivalent to another self-similar set with the strong separation condtion. In [42], Tian et 2 . This result is sharp.…”

“…Very useful is the following lemma. It comes from [3] and [42]. For the convenience of readers, we give the detailed proof.…”