PreprintCC BY 4.0arXiv
On a problem of Ma and Chen concerning minimal additive complements
Abstract
Let m≥1 be an integer and let W=(mℕ0+X)∪ U∪ V⊆ℤ be bounded below, where X≠∅, [U]m⊆[X]m, [X]m∩[V]m=∅, and V contains no infinite arithmetic progression with common difference m. Here [S]m denotes the set of residue classes modulo m represented by S. Suppose that W satisfies the 𝒞-condition modulo m. For 0≤ r<m, let Vr=V∩(r+mℤ), and put V'=⋃|Vr|=∞Vr, where the union is over 0≤ r<m. We prove that W has a minimal additive complement if V' satisfies the big-hole property modulo m. This answers a problem posed by Ma and Chen.
Related
BibTeX
@misc{chen2026problem,
title = {{On a problem of Ma and Chen concerning minimal additive complements}},
author = {Shi-Qiang Chen and Quan-Hui Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26608},
eprint = {2610.26608},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26608},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a problem of Ma and Chen concerning minimal additive complements
AU - Shi-Qiang Chen
AU - Quan-Hui Yang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26608
UR - https://arxiv.org/abs/2610.26608
AB - Let m≥1 be an integer and let W=(mℕ_0+X)∪ U∪ V⊆ℤ be bounded below, where X≠∅, [U]_m⊆[X]_m, [X]_m∩[V]_m=∅, and V contains no infinite arithmetic progression with common difference m. Here [S]_m denotes the set of residue classes modulo m represented by S. Suppose that W satisfies the 𝒞-condition modulo m. For 0≤ r<m, let V_r=V∩(r+mℤ), and put V'=⋃_|V_r|=∞V_r, where the union is over 0≤ r<m. We prove that W has a minimal additive complement if V' satisfies the big-hole property modulo m. This answers a problem posed by Ma and Chen.
ER -