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

2025/12/19 by Piotr Nowakowski, Nowakowski, Piotr
Mathematics · #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals #Advanced Banach Space Theory

paper · doi:10.48550/arxiv.2512.17761

Abstract

Given a nonincreasing sequence of positive numbers (a_n) such that the series ∑ a_n is convergent, by E(a_n) we denote the set of all subsums of the series ∑ 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.

Citations

• Kenyon Theorem Revisited
• Achievable Cantorvals almost without reversed Kakeya conditions
• Algebraic sums of achievable sets involving Cantorvals
• Conditions for the difference set of a central Cantor set to be a Cantorval. Part II
• Set of uniqueness for Cantorvals
• Multigeometric sequences and Cantorvals
• Projecting the one-dimensional Sierpinski gasket

Related

• Algebraic sums of achievable sets involving Cantorvals
• Multigeometric sequences and Cantorvals
• Achievement sets of conditionally convergent series
• Achievement sets and sum ranges with ideal supports
• Subsums of conditionally convergent series in finite dimensional spaces