PreprintarXiv licencearXiv
Sets of cardinality seven are not sum-dominant
Abstract
We give a self-contained elementary proof of the known result that every set A⊂R of cardinality seven satisfies |A+A|≤ |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.
Related
- Quantum Latin Hypercubes with Maximal Cardinality
- Symmetric Quantum Latin Squares with Maximal Cardinality
- Orthogonal Quantum Latin Squares with Maximal Cardinality from Full-Rank Factorizations
- Resolvable quantum Latin squares with maximal cardinality
- On the cardinality of the exception set in Littlewood's discrete conjecture
BibTeX
@misc{do2026sets,
title = {{Sets of cardinality seven are not sum-dominant}},
author = {Thai Duong Do and Van Thien Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36415},
eprint = {2609.36415},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36415},
}
RIS
TY - UNPB
TI - Sets of cardinality seven are not sum-dominant
AU - Thai Duong Do
AU - Van Thien Nguyen
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36415
UR - https://arxiv.org/abs/2609.36415
AB - We give a self-contained elementary proof of the known result that every set A⊂R of cardinality seven satisfies |A+A|≤ |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.
ER -