On subsets of asymptotic bases

Ji-Zhen Xu; Yong-Gao Chen

doi:10.5802/crmath.513

Article de recherche - Théorie des nombres

On subsets of asymptotic bases

Ji-Zhen Xu ^1,² ; Yong-Gao Chen 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

Comptes Rendus. Mathématique, Volume 362 (2024), 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 :

Ji-Zhen Xu ^{1, 2} ; Yong-Gao Chen ¹

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2024__362_G1_45_0,
     author = {Ji-Zhen Xu and Yong-Gao Chen},
     title = {On subsets of asymptotic bases},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {45--49},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.513},
     language = {en},
}

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Ji-Zhen Xu; Yong-Gao Chen. On subsets of asymptotic bases. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 45-49. doi : 10.5802/crmath.513. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.513/

Bibliographie
Cité par

[1] Feng-Juan Chen; Yong-Gao Chen On minimal asymptotic bases, Eur. J. Comb., Volume 32 (2011) no. 8, pp. 1329-1335 | DOI | MR | Zbl

[2] Yong-Gao Chen; Min Tang On a problem of Nathanson, Acta Arith., Volume 185 (2018) no. 3, pp. 275-280 | DOI | MR | Zbl

[3] Paul Erdős; Melvyn B. Nathanson Minimal asymptotic bases with prescribed densities, Ill. J. Math., Volume 32 (1988) no. 3, pp. 562-574 | MR | Zbl

[4] Erich Härtter Ein Beitrag zur Theorie der Minimalbasen, J. Reine Angew. Math., Volume 196 (1956), pp. 170-204 | DOI | MR | Zbl

[5] Miroslawa Jańczak; Tomasz Schoen Dense minimal asymptotic bases of order $2$ , J. Number Theory, Volume 130 (2010) no. 3, pp. 580-585 | DOI | MR | Zbl

[6] Martin Kneser Abschätzung der asymptotischen Dichte von Summenmengen, Math. Z., Volume 58 (1953), pp. 459-484 | DOI | Zbl

[7] Melvyn B. Nathanson Minimal bases and maximal nonbases in additive number theory, J. Number Theory, Volume 6 (1974), pp. 324-333 | DOI | MR | Zbl

[8] Melvyn B. Nathanson Minimal bases and powers of $2$ , Acta Arith., Volume 51 (1988) no. 5, pp. 95-102

[9] Melvyn B. Nathanson; András Sárközy On the maximum density of minimal asymptotic bases, Proc. Am. Math. Soc., Volume 105 (1989) no. 1, pp. 31-33 | DOI | MR | Zbl

[10] Alfred Stöhr Gelöste und ungelöste Fragen über Basen der natüurlichen Zahlenreihe, J. Reine Angew. Math., Volume 194 (1955), pp. 111-140 | DOI | Zbl

[11] Cui-Fang Sun On a problem of Nathanson on minimal asymptotic bases, J. Number Theory, Volume 218 (2021), pp. 152-160 | MR | Zbl

[12] Min Tang; Deng-Rong Ling On asymptotic bases and minimal asymptotic bases, Colloq. Math., Volume 170 (2022) no. 1, pp. 65-77 | DOI | MR | Zbl

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