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arXiv · 2609.36415

Sets of cardinality seven are not sum-dominant

Thai Duong Do·Van Thien Nguyen
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Abstract

We give a self-contained elementary proof of the known result that every set $A\subset\R$ of cardinality seven satisfies $|A+A|\le |A-A|$. The argument adapts Hegarty's method, using representation counts and the largest positive differences. An exact counting identity, two applications of the Cauchy--Schwarz inequality, and an analysis of a symmetric six-element set with one point added complete the proof, with no computer enumeration. This answers in the affirmative a question of Chu for the seven-element case.

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Thai Duong Do, Van Thien Nguyen. 2026-09-29. Sets of cardinality seven are not sum-dominant. https://arxiv.org/abs/2609.36415

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