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PreprintCC BY 4.0arXiv

On a problem of Nathanson related to minimal asymptotic bases and maximal asymptotic nonbases

  • Shi-Qiang Chen
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Abstract

Let ℕ0={0,1,2,…} and let h≥2 be an integer. For a set A⊆ℕ0, write hA for the set of all sums of h, not necessarily distinct, elements of A. In this paper, we prove that for every h≥2, there is a partition ℕ0=Asqcup B such that A is a minimal asymptotic basis of order h and B is a maximal asymptotic nonbasis of order h. This solves an open problem posed by Nathanson in 1974.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026problem,
  title = {{On a problem of Nathanson related to minimal asymptotic bases and maximal asymptotic nonbases}},
  author = {Shi-Qiang Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07361},
  eprint = {2609.07361},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07361},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On a problem of Nathanson related to minimal asymptotic bases and maximal asymptotic nonbases
AU  - Shi-Qiang Chen
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07361
UR  - https://arxiv.org/abs/2609.07361
AB  - Let ℕ_0={0,1,2,…} and let h≥2 be an integer. For a set A⊆ℕ_0, write hA for the set of all sums of h, not necessarily distinct, elements of A. In this paper, we prove that for every h≥2, there is a partition ℕ_0=Asqcup B such that A is a minimal asymptotic basis of order h and B is a maximal asymptotic nonbasis of order h. This solves an open problem posed by Nathanson in 1974.
ER  -