On a problem of Nathanson related to minimal asymptotic bases of order $h$

Shi-Qiang Chen; Csaba Sándor; Quan-Hui Yang

doi:10.5802/crmath.530

Research article - Number theory

On a problem of Nathanson related to minimal asymptotic bases of order

h

Shi-Qiang Chen ¹; Csaba Sándor ^2,^3,⁴; Quan-Hui Yang ⁵

¹ School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P. R. China
² Department of Stochastics, Institute of Mathematics, BudapestUniversity of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
³ Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
⁴ MTA-BME Lendület Arithmetic Combinatorics Research Group, ELKH, Műegyetem rkp. 3., H-1111 Budapest, Hungary
⁵ School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 71-76.

Abstract

For integer $h \geq 2$ and $A \subseteq ℕ$ , we define $h A$ to be the set of all integers which can be written as a sum of $h$ , not necessarily distinct, elements of $A$ . The set $A$ is called an asymptotic basis of order $h$ if $n \in h A$ for all sufficiently large integers $n$ . An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$ . For $W \subseteq ℕ$ , denote by $ℱ^{*} (W)$ the set of all finite, nonempty subsets of $W$ . Let $A (W)$ be the set of all numbers of the form $\sum_{f \in F} 2^{f}$ , where $F \in ℱ^{*} (W)$ . In this paper, we give some characterizations of the partitions $ℕ = W_{1} \cup \dots \cup W_{h}$ with the property that $A = A (W_{1}) \cup \dots \cup A (W_{h})$ is a minimal asymptotic basis of order $h$ . This generalizes a result of Chen and Chen, recent result of Ling and Tang, and also recent result of Sun.

Received: 2023-02-15
Revised: 2023-06-12
Accepted: 2023-06-26
Published online: 2024-02-02

DOI: 10.5802/crmath.530

Classification: 11B34
Keywords: Asymptotic bases, minimal asymptotic bases, binary representation

Author's affiliations:

Shi-Qiang Chen ¹; Csaba Sándor ^{2, 3, 4}; Quan-Hui Yang ⁵

¹ School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P. R. China
² Department of Stochastics, Institute of Mathematics, BudapestUniversity of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
³ Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
⁴ MTA-BME Lendület Arithmetic Combinatorics Research Group, ELKH, Műegyetem rkp. 3., H-1111 Budapest, Hungary
⁵ School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2024__362_G1_71_0,
     author = {Shi-Qiang Chen and Csaba S\'andor and Quan-Hui Yang},
     title = {On a problem of {Nathanson} related to minimal asymptotic bases of order $h$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {71--76},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.530},
     language = {en},
}

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Shi-Qiang Chen; Csaba Sándor; Quan-Hui Yang. On a problem of Nathanson related to minimal asymptotic bases of order $h$. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 71-76. doi : 10.5802/crmath.530. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.530/

References
Cited by

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