When the algebraic difference of two central Cantor sets is an interval?

Abstract

Let C(a), C(b) ⊂ [0, 1] be the central Cantor sets generated by sequences a, b ∈ (0, 1) N . The main result in the first part of the paper gives a necessary condition and a sufficient condition for sequences a and b which inform when C(a) − C(b) is equal to [−1, 1] or is a finite union of closed intervals. In the second part we investigate some self-similar Cantor sets C(l, r, p), which we call S-Cantor sets, generated by numbers l, r, p ∈ N, l + r < p. We give a full characterization of the set C(l1, r1, p) − C(l2, r2, p) which can take one of the form: the interval [−1, 1], a Cantor set, an L-Cantorval, an R-Cantorval or an M-Cantorval.