Journal of Integer Sequences, Vol. 22 (2019), Article 19.3.7

## When Sets Are Not Sum-Dominant

### Hùng Việt Chu Department of Mathematics Washington and Lee University Lexington, VA 24450 USA

Abstract:

Given a set A of nonnegative integers, define the sum set A + A = {ai + aj | ai, ajA} and the difference set AA = {aiaj | ai, ajA}. The set A is said to be sum-dominant if |A + A| > |AA|. 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.

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Received March 6 2019; revised versions received May 17 2019; May 20 2019. Published in Journal of Integer Sequences, May 21 2019.