Multigeometric sequences and Cantorvals

TL;DR

This paper classifies the achievement sets of certain sequences, showing that they can be finite unions of intervals, Cantor sets, or Cantorvals, and describes families of sequences producing Cantorvals.

Contribution

It introduces new families of sequences whose achievement sets are Cantorvals, expanding the understanding of the structure of these sets.

Findings

Achievement sets can be finite unions, Cantor sets, or Cantorvals.

Identifies all known examples of sequences with Cantorval achievement sets.

Provides a framework to generate sequences with Cantorval achievement sets.

Abstract

For a sequence $x \in l_1 \setminus c_{00}$, one can consider the achievement set $E(x)$ of all subsums of series $\sum_{n=1}^{\infty} x(n)$. It is known that $E(x)$ is one of the following types of sets: * finite union of closed intervals, * homeomorphic to the Cantor set, * homeomorphic to the set $T$ of subsums of $\sum_{n=1}^{\infty} c(n)$ where $c(2n-1)=\frac{3}{4^n}$ and $c(2n)=\frac{2}{4^n}$ (Cantorval). Based on ideas of Jones and Velleman, and Guthrie and Nymann we describe families of sequences which contain, according to our knowledge, all known examples of $x$'s with $E(x)$ being Cantorvals.