On a problem of Nathanson related to minimal asymptotic bases of order $h$
For integer $h\ge 2$ and $A\subseteq \mathbb{N}$, 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 b...
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Académie des sciences
2024-02-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.530/ |
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| author | Chen, Shi-Qiang Sándor, Csaba Yang, Quan-Hui |
| author_facet | Chen, Shi-Qiang Sándor, Csaba Yang, Quan-Hui |
| author_sort | Chen, Shi-Qiang |
| collection | DOAJ |
| description | For integer $h\ge 2$ and $A\subseteq \mathbb{N}$, 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 $W\subseteq \mathbb{N}$, denote by $\mathcal{F}^*(W)$ 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 $F \in \mathcal{F}^*(W)$. In this paper, we give some characterizations of the partitions $\mathbb{N}=W_1\cup \dots \cup W_h$ 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. |
| format | Article |
| id | doaj-art-9c93450d06a14de7b3e6b871169e00d9 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-02-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-9c93450d06a14de7b3e6b871169e00d92025-02-07T11:12:53ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-02-01362G1717610.5802/crmath.53010.5802/crmath.530On a problem of Nathanson related to minimal asymptotic bases of order $h$Chen, Shi-Qiang0Sándor, Csaba1Yang, Quan-Hui2School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P. R. ChinaDepartment of Computer Science and Information Theory, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary; Department of Stochastics, Institute of Mathematics, BudapestUniversity of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary; MTA-BME Lendület Arithmetic Combinatorics Research Group, ELKH, Műegyetem rkp. 3., H-1111 Budapest, HungarySchool of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, ChinaFor integer $h\ge 2$ and $A\subseteq \mathbb{N}$, 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 $W\subseteq \mathbb{N}$, denote by $\mathcal{F}^*(W)$ 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 $F \in \mathcal{F}^*(W)$. In this paper, we give some characterizations of the partitions $\mathbb{N}=W_1\cup \dots \cup W_h$ 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.530/Asymptotic basesminimal asymptotic basesbinary representation |
| spellingShingle | Chen, Shi-Qiang Sándor, Csaba Yang, Quan-Hui On a problem of Nathanson related to minimal asymptotic bases of order $h$ Comptes Rendus. Mathématique Asymptotic bases minimal asymptotic bases binary representation |
| title | On a problem of Nathanson related to minimal asymptotic bases of order $h$ |
| title_full | On a problem of Nathanson related to minimal asymptotic bases of order $h$ |
| title_fullStr | On a problem of Nathanson related to minimal asymptotic bases of order $h$ |
| title_full_unstemmed | On a problem of Nathanson related to minimal asymptotic bases of order $h$ |
| title_short | On a problem of Nathanson related to minimal asymptotic bases of order $h$ |
| title_sort | on a problem of nathanson related to minimal asymptotic bases of order h |
| topic | Asymptotic bases minimal asymptotic bases binary representation |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.530/ |
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