On a problem of Nathanson related to minimal asymptotic bases of order $h$

Shi-Qiang Chen; Csaba Sándor; Quan-Hui Yang

doi:10.5802/crmath.530

Article de recherche - Théorie des nombres

On a problem of Nathanson related to minimal asymptotic bases of order

h

Shi-Qiang Chen ¹ ; Csaba Sándor ^2,^3,⁴ ; Quan-Hui Yang ⁵

¹ School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P. R. China
² Department of Stochastics, Institute of Mathematics, BudapestUniversity of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
³ Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
⁴ MTA-BME Lendület Arithmetic Combinatorics Research Group, ELKH, Műegyetem rkp. 3., H-1111 Budapest, Hungary
⁵ School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 71-76.

Résumé

For integer $h \geq 2$ and $A \subseteq ℕ$ , we define $h A$ 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 $n \in h A$ 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 $W \subseteq ℕ$ , denote by $ℱ^{*} (W)$ 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 $F \in ℱ^{*} (W)$ . In this paper, we give some characterizations of the partitions $ℕ = W_{1} \cup \dots \cup W_{h}$ 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.

Reçu le : 2023-02-15
Révisé le : 2023-06-12
Accepté le : 2023-06-26
Publié le : 2024-02-02

DOI : 10.5802/crmath.530

Classification : 11B34
Mots clés : Asymptotic bases, minimal asymptotic bases, binary representation

Affiliations des auteurs :

Shi-Qiang Chen ¹ ; Csaba Sándor ^{2, 3, 4} ; Quan-Hui Yang ⁵

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Shi-Qiang Chen and Csaba S\'andor and Quan-Hui Yang},
     title = {On a problem of {Nathanson} related to minimal asymptotic bases of order $h$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {71--76},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.530},
     language = {en},
}

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Shi-Qiang Chen; Csaba Sándor; Quan-Hui Yang. On a problem of Nathanson related to minimal asymptotic bases of order $h$. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 71-76. doi : 10.5802/crmath.530. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.530/

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 for the natural numbers, J. Number Theory, Volume 12 (1980), pp. 154-159 | DOI | 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 two, J. Number Theory, Volume 130 (2010) no. 3, pp. 580-585 | DOI | MR | Zbl

[6] Xingde Jia; Melvyn B. Nathanson A simple construction of minimal asymptotic bases, Acta Arith., Volume 52 no. 2, pp. 95-101 | MR | Zbl

[7] Deng-Rong Ling; Min Tang Some remarks on minimal asymptotic bases of order three, Bull. Aust. Math. Soc., Volume 102 (2020) no. 1, pp. 21-30 | DOI | MR | Zbl

[8] Melvyn B. Nathanson Minimal bases and powers of $2$ , Acta Arith., Volume 49 (1988) no. 5, pp. 525-532 | DOI | MR | Zbl

[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 natürlichen Zahlenreihe II, J. Reine Angew. Math., Volume 194 (1955), pp. 40-65 | DOI

[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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