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

Abstract

Let $$C(\lambda )\subset [0,1]$$ C ( λ ) ⊂ [ 0 , 1 ] denote the central Cantor set generated by a sequence $$\lambda =(\lambda _{n})\in \left( 0,\frac{1}{2} \right) ^{\mathbb {N}}$$ λ = ( λ n ) ∈ 0 , 1 2 N . By the known trichotomy, the difference set $$ C(\lambda )-C(\lambda )$$ C ( λ ) - C ( λ ) of $$C(\lambda )$$ C ( λ ) 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 $$(\lambda _{n})$$ ( λ n ) which guarantee that $$C(\lambda )-C(\lambda )$$ C ( λ ) - C ( λ ) is a Cantorval. We show that these conditions can be expressed in several equivalent forms. Under additional assumptions, the measure of the Cantorval $$C(\lambda )-C(\lambda )$$ C ( λ ) - C ( λ ) is established. We give an application of the proved theorems for the achievement sets of some fast convergent series.