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

Abstract

Let $C\left(a\right)\subset [0,1]$ be the central Cantor set generated by a sequence $a=\left(a_n\right)\in {\left(0,1\right)}^N$. It is known that the difference set $C\left(a\right)-C\left(a\right)$ has one of three possible forms: a finite union of closed intervals, a Cantor set, or a Cantorval. In the previous paper (Filipczak and Nowakowski, 2023), there was given the sufficient condition for the sequence $a$, which implies that $C\left(a\right)-C\left(a\right)$ is a Cantorval. In this paper we give different conditions for a sequence $a$, which guarantee the same assertion. We also prove a corollary, which provides infinitely many new examples of Cantorvals.