# When Sets Are Not Sum-dominant

## TL;DR

The paper provides a human-understandable proof that sets with fewer than six elements are not sum-dominant and explores how adding elements to arithmetic progressions or specific sequences affects sum-dominance.

## Contribution

It offers a human-proof that sets smaller than six are not sum-dominant and analyzes how augmenting sets with certain elements influences sum-dominance.

## Key findings

- Sets with fewer than six elements are not sum-dominant.
- Adding up to two elements to an arithmetic progression does not produce a sum-dominant set.
- Incorporating specific sequence elements with a few arbitrary numbers does not yield sum-dominance.

## Abstract

Given a set $A$ of nonnegative integers, define the sum set $$A+A = \{a_i+a_j\mid a_i,a_j\in A\}$$ and the difference set $$A-A = \{a_i-a_j\mid a_i,a_j\in A\}.$$ The set $A$ is said to be sum-dominant if $|A+A|>|A-A|$. In answering a question by Nathanson, Hegarty used a clever algorithm to find that the smallest cardinality of a sum-dominant set is $8$. Since then, Nathanson has been asking for a human-understandable proof of the result. We offer a computer-free proof that a set of cardinality less than $6$ is not sum-dominant. Furthermore, we prove that the introduction of at most two numbers into a set of numbers in an arithmetic progression does not give a sum-dominant set. This theorem eases several of our proofs and may shed light on future work exploring why a set of cardinality $6$ is not sum-dominant. Finally, we prove that if a set contains a certain number of integers from a specific sequence, then adding a few arbitrary numbers into the set does not give a sum-dominant set.