no. G1
Article de recherche - Théorie des nombres
On a problem of Nathanson related to minimal asymptotic bases of order h
Chen, Shi-Qiang1 ; Sándor, Csaba2, 3, 4 ; Yang, Quan-Hui5 
1School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P. R. China
2Department of Stochastics, Institute of Mathematics, BudapestUniversity of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
3Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
4MTA-BME Lendület Arithmetic Combinatorics Research Group, ELKH, Műegyetem rkp. 3., H-1111 Budapest, Hungary
5School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Comptes Rendus. Mathématique, Tome 362 (2024) no. G1, pp. 71-76

For integer h≥2 and A⊆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∈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⊆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 ∑ f∈F 2 f , where F∈ℱ * (W). In this paper, we give some characterizations of the partitions ℕ=W 1 ∪⋯∪W h with the property that A=A(W 1 )∪⋯∪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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DOI : 10.5802/crmath.530
Classification : 11B34
Keywords: Asymptotic bases, minimal asymptotic bases, binary representation

Chen, Shi-Qiang  1   ; Sándor, Csaba  2 , 3 , 4   ; Yang, Quan-Hui  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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {On a problem of {Nathanson} related to minimal asymptotic bases of order $h$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {71--76},
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Chen, Shi-Qiang; Sándor, Csaba; Yang, Quan-Hui. On a problem of Nathanson related to minimal asymptotic bases of order $h$. Comptes Rendus. Mathématique, Tome 362 (2024) no. G1, pp. 71-76. doi: 10.5802/crmath.530

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