Subsum Sets: Intervals, Cantor Sets, and Cantorvals

TL;DR

This paper explores the structure of subsum sets derived from sequences converging to zero, classifying them into intervals, Cantor sets, or Cantorvals based on summability properties.

Contribution

It provides a comprehensive classification of subsum sets for sequences converging to zero, linking their structure to the summability of the original sequence.

Findings

Subsum sets are unbounded intervals if the sequence is not absolutely summable.

Absolutely summable sequences produce subsum sets that are finite unions of intervals, Cantor sets, or Cantorvals.

The paper characterizes the conditions under which each type of subsum set occurs.

Abstract

Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which includes zero. When it is absolutely summable the subsum set is one of the following: a finite union of (nontrivial) compact intervals, a Cantor set, or a "symmetric Cantorval".