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

Abstract

Let $C(\lambda )\subset [0,1]$ 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.

19 pages

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