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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-02-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.530/ |
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| Summary: | 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. |
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| ISSN: | 1778-3569 |