Conditions for the Difference Set of a Central Cantor Set to be a Cantorval

Let C(?) ? [0, 1] denote the central Cantor set generated by a sequence ? = (?(n)) ? (0, 1/2)(N). By the known trichotomy, the difference set 2 C(?) - C(?) of C(?) is one of three possible sets: a finite union of closed intervals, a Cantor set, or a Cantorval. Our main result describes effective conditions for (?(n)) which guarantee that C(?) - C(?) is a Cantorval. We show that these conditions can be expressed in several equivalent forms. Under additional assumptions, the measure of the Cantorval C(?) -C(?) is established. We give an application of the proved theorems for the achievement sets of some fast convergent series.