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PreprintCC BY 4.0arXiv

On strongly minimal asymptotic bases of order h generated by powers of $2$

  • Shi-Qiang Chen
  • Xin-Yue Yan
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Abstract

Let h≥2 be an integer and let ℕ0=W1∪⋯∪ Wh be a partition into infinite sets. For W⊆ℕ0, let A(W) denote the set of all finite nonempty sums of distinct powers 2w with w∈ W. We give a necessary and sufficient condition for A=A(W1)∪⋯∪ A(Wh) to be a strongly minimal asymptotic basis of order h. This provides a complete answer to a problem posed by Nathanson in 1988.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026strongly,
  title = {{On strongly minimal asymptotic bases of order $h$ generated by powers of $2$}},
  author = {Shi-Qiang Chen and Xin-Yue Yan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15416},
  eprint = {2610.15416},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15416},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On strongly minimal asymptotic bases of order h generated by powers of $2$
AU  - Shi-Qiang Chen
AU  - Xin-Yue Yan
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15416
UR  - https://arxiv.org/abs/2610.15416
AB  - Let h≥2 be an integer and let ℕ_0=W_1∪⋯∪ W_h be a partition into infinite sets. For W⊆ℕ_0, let A(W) denote the set of all finite nonempty sums of distinct powers 2^w with w∈ W. We give a necessary and sufficient condition for A=A(W_1)∪⋯∪ A(W_h) to be a strongly minimal asymptotic basis of order h. This provides a complete answer to a problem posed by Nathanson in 1988.
ER  -