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

TL;DR

This paper explores conditions under which the difference set of a central Cantor set forms a Cantorval, providing new criteria and infinitely many examples of such sets.

Contribution

It introduces new conditions on the generating sequence that guarantee the difference set is a Cantorval, expanding the understanding of their structure.

Findings

New sufficient conditions for difference sets to be Cantorvals

Infinite new examples of Cantorvals

Extension of previous criteria to broader cases

Abstract

Let C(a) be the central Cantor set generated by a sequence a with terms in (0,1). It is known that the difference set C(a)-C(a) of C(a) can has one of three possible forms: a finite union of closed intervals, a Cantor set, or a Cantorval. In the previous paper there was proved a sufficient condition for the sequence a which implies that C(a) - C(a) 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.