On subsets of asymptotic bases
Let $h\ge 2$ 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 $\varepsilon >0$, there exists a finite...
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Académie des sciences
2024-02-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.513/ |
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| author | Xu, Ji-Zhen Chen, Yong-Gao |
| author_facet | Xu, Ji-Zhen Chen, Yong-Gao |
| author_sort | Xu, Ji-Zhen |
| collection | DOAJ |
| description | Let $h\ge 2$ 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 $\varepsilon >0$, there exists a finite subset $F_\varepsilon $ of $A$ such that $d_L(h(F_\varepsilon \cup B))\ge hd_L(B)-\varepsilon $, where $d_L(X)$ denotes the lower asymptotic density of $X$ and $hX$ denotes the set of all $x_1+\cdots +x_h$ with $x_i\in X$ $(1\le i\le h)$. This generalizes a result of Nathanson and Sárközy. |
| format | Article |
| id | doaj-art-3af80b6ce2634f04b4c1cdc88dc3e5c6 |
| 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-3af80b6ce2634f04b4c1cdc88dc3e5c62025-02-07T11:12:53ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-02-01362G1454910.5802/crmath.51310.5802/crmath.513On subsets of asymptotic basesXu, Ji-Zhen0Chen, Yong-Gao1Nanjing Vocational College of Information Technology,Nanjing 210023, People’s Republic of China; School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, People’s Republic of ChinaSchool of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, People’s Republic of ChinaLet $h\ge 2$ 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 $\varepsilon >0$, there exists a finite subset $F_\varepsilon $ of $A$ such that $d_L(h(F_\varepsilon \cup B))\ge hd_L(B)-\varepsilon $, where $d_L(X)$ denotes the lower asymptotic density of $X$ and $hX$ denotes the set of all $x_1+\cdots +x_h$ with $x_i\in X$ $(1\le i\le h)$. This generalizes a result of Nathanson and Sárközy.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.513/ |
| spellingShingle | Xu, Ji-Zhen Chen, Yong-Gao On subsets of asymptotic bases Comptes Rendus. Mathématique |
| title | On subsets of asymptotic bases |
| title_full | On subsets of asymptotic bases |
| title_fullStr | On subsets of asymptotic bases |
| title_full_unstemmed | On subsets of asymptotic bases |
| title_short | On subsets of asymptotic bases |
| title_sort | on subsets of asymptotic bases |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.513/ |
| work_keys_str_mv | AT xujizhen onsubsetsofasymptoticbases AT chenyonggao onsubsetsofasymptoticbases |