PreprintCC BY 4.0arXiv
Inverse Zero-Sum Problems of Erdős--Ginzburg--Ziv constant: Property C and Property D conjectures
Abstract
Let Cnr denote the direct sum of r cyclic groups of order n, where n≥2 and r≥1. Properties C and D predict a uniform multiplicity structure for extremal zero-sum problems over Cnr: every sequence of maximum length avoiding nonempty zero-sum subsequences of length at most n, or of length exactly n, respectively, should have the form Tn−1. We prove that, for every r≥4 and every even n≥2, each of these properties holds if and only if n is a power of two. This gives negative answers to the general Property C problem posed by Gao, Hou, Schmid and Thangadurai Open Problems 1.2{GHST} in 2007, and disproves the Property D conjecture proposed by Gao and Geroldinger [Conjecture 7.2]{GG} in 2006.
Related
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BibTeX
@misc{wang2026inverse,
title = {{Inverse Zero-Sum Problems of Erdős--Ginzburg--Ziv constant: Property C and Property D conjectures}},
author = {Guoqing Wang and Xiangneng Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25271},
eprint = {2610.25271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25271},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Inverse Zero-Sum Problems of Erdős--Ginzburg--Ziv constant: Property C and Property D conjectures
AU - Guoqing Wang
AU - Xiangneng Zeng
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.25271
UR - https://arxiv.org/abs/2610.25271
AB - Let C_n^r denote the direct sum of r cyclic groups of order n, where n≥2 and r≥1. Properties C and D predict a uniform multiplicity structure for extremal zero-sum problems over C_n^r: every sequence of maximum length avoiding nonempty zero-sum subsequences of length at most n, or of length exactly n, respectively, should have the form T^n−1. We prove that, for every r≥4 and every even n≥2, each of these properties holds if and only if n is a power of two. This gives negative answers to the general Property C problem posed by Gao, Hou, Schmid and Thangadurai Open Problems 1.2{GHST} in 2007, and disproves the Property D conjecture proposed by Gao and Geroldinger [Conjecture 7.2]{GG} in 2006.
ER -