On subsets of asymptotic bases

Xu, Ji-Zhen; Chen, Yong-Gao

doi:10.5802/crmath.513

Article de recherche - Théorie des nombres

On subsets of asymptotic bases

¹School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, People’s Republic of China
²Nanjing Vocational College of Information Technology,Nanjing 210023, People’s Republic of China

Comptes Rendus. Mathématique, Tome 362 (2024) no. G1, pp. 45-49

Résumé

Let $h \geq 2$ be an integer. In this paper, we prove that if $A$ is an asymptotic basis of order $h$ and $B$ is a nonempty subset of $A$ , then either there exists a finite subset $F$ of $A$ such that $F \cup B$ is an asymptotic basis of order $h$ , or for any $ε > 0$ , there exists a finite subset $F_{ε}$ of $A$ such that $d_{L} (h (F_{ε} \cup B)) \geq h d_{L} (B) - ε$ , where $d_{L} (X)$ denotes the lower asymptotic density of $X$ and $h X$ denotes the set of all $x_{1} + \dots + x_{h}$ with $x_{i} \in X$ $(1 \leq i \leq h)$ . This generalizes a result of Nathanson and Sárközy.

Reçu le : 2023-01-25
Accepté le : 2023-05-13
Publié le : 2024-02-02

DOI : 10.5802/crmath.513

Classification : 11B13, 11B05, 11P99

Affiliations des auteurs :

Xu, Ji-Zhen ^{1
,

2} ; Chen, Yong-Gao ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, People’s Republic of China
² Nanjing Vocational College of Information Technology,Nanjing 210023, People’s Republic of China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On subsets of asymptotic bases},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {45--49},
     year = {2024},
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     volume = {362},
     number = {G1},
     doi = {10.5802/crmath.513},
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     url = {https://www.numdam.org/articles/10.5802/crmath.513/}
}

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Xu, Ji-Zhen; Chen, Yong-Gao. On subsets of asymptotic bases. Comptes Rendus. Mathématique, Tome 362 (2024) no. G1, pp. 45-49. doi: 10.5802/crmath.513

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