Research article - Number theory
On a problem of Nathanson related to minimal asymptotic bases of order $h$
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 71-76.

For integer $h\ge 2$ and $A\subseteq ℕ$, we define $hA$ 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 $n\in hA$ 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 ${ℱ}^{*}\left(W\right)$ the set of all finite, nonempty subsets of $W$. Let $A\left(W\right)$ be the set of all numbers of the form ${\sum }_{f\in F}{2}^{f}$, where $F\in {ℱ}^{*}\left(W\right)$. In this paper, we give some characterizations of the partitions $ℕ={W}_{1}\cup \cdots \cup {W}_{h}$ with the property that $A=A\left({W}_{1}\right)\cup \cdots \cup A\left({W}_{h}\right)$ 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.530
Classification: 11B34
Keywords: Asymptotic bases, minimal asymptotic bases, binary representation

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
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title = {On a problem of {Nathanson} related to minimal asymptotic bases of order $h$},
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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/

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