Conditions for the difference set of a central Cantor set to be a Cantorval

Abstract

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