On a new condition implying that an achievement set is a Cantorval and its applications

TL;DR

The paper addresses when a convergent-sum achievement set E(a_n) is a Cantorval, extending the historical Kakeya dichotomy with a new sufficient condition. It introduces the Star Procedure and a detailed gap-coverage framework that guarantees Cantorval structure when the procedure can be continued indefinitely. The authors provide explicit multigeometric sequence constructions yielding Cantorvals, including new examples outside previously known parameter ranges, and connect these results to broader questions about Cantor sets, Cantorvals, and Palis-type conjectures. Overall, the work offers a practical criterion for generating Cantorvals and advances understanding of the topological forms that achievement sets can assume.

Abstract

Given a nonincreasing sequence of positive numbers $(a_n)$ such that the series $\sum a_n$ is convergent, by $E(a_n)$ we denote the set of all subsums of the series $\sum a_n$ and call it the achievement set of $(a_n)$. It is well known that such a set can be a finite union of closed intervals, a Cantor set or a Cantorval. We give a new condition implying that the last possibility occurs. We also show how we can use this condition to produce new achievable Cantorvals. In particular, we prove that Kakeya conditions cannot tell us more about the form of the achievement set than it was proved by Kakeya.

Figures (4)

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Theorems & Definitions (12)

• Definition 1
• Lemma 2.1
• Definition 2
• Theorem 3.1
• proof
• Example 3
• Theorem 4.1
• proof
• Example 4
• Example 5