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

TL;DR

This paper introduces a new condition that guarantees the achievement set of a convergent series of positive, nonincreasing terms is a Cantorval, and demonstrates how to generate new examples of such sets.

Contribution

The paper presents a novel condition for achievement sets to be Cantorvals and explores its applications in constructing new examples of these sets.

Findings

A new condition for achievement sets to be Cantorvals.

Construction of new Cantorval achievement sets.

Kakeya conditions do not provide additional insights into achievement set structures.

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.