Generalized H-fold sumset and Subsequence sum

Mohan; Ram Krishna Pandey

doi:10.5802/crmath.483

Article de recherche - Combinatoire, Théorie des nombres

Generalized H-fold sumset and Subsequence sum

Mohan ¹ ; Ram Krishna Pandey ²

¹ Department of Mathematics, Indian Institute of Technology Roorkee, Uttarakhand 247667, India
² Department of Mathematics, Indian Institute of Technology Roorkee, Uttarakhand, 247667, India

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1-19.

Résumé

Let $A$ and $H$ be nonempty finite sets of integers and positive integers, respectively. The generalized $H$ -fold sumset, denoted by $H^{(r)} A$ , is the union of the sumsets $h^{(r)} A$ for $h \in H$ where, the sumset $h^{(r)} A$ is the set of all integers that can be represented as a sum of $h$ elements from $A$ with no summand in the representation appearing more than $r$ times. In this paper, we find the optimal lower bound for the cardinality of $H^{(r)} A$ , i.e., for $| H^{(r)} A |$ and the structure of the underlying sets $A$ and $H$ when $| H^{(r)} A |$ is equal to the optimal lower bound in the cases $A$ contains only positive integers and $A$ contains only nonnegative integers. This generalizes recent results of Bhanja. Furthermore, with a particular set $H$ , since $H^{(r)} A$ generalizes subsequence sum and hence subset sum, we get several results of subsequence sums and subset sums as special cases.

Reçu le : 2022-09-07
Révisé le : 2023-02-21
Accepté le : 2023-02-23
Publié le : 2024-02-02

DOI : 10.5802/crmath.483

Classification : 11P70, 11B75, 11B13
Mots clés : sumset, subset sum, subsequence sum

Affiliations des auteurs :

Mohan ¹ ; Ram Krishna Pandey ²

¹ Department of Mathematics, Indian Institute of Technology Roorkee, Uttarakhand 247667, India
² Department of Mathematics, Indian Institute of Technology Roorkee, Uttarakhand, 247667, India

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Generalized {H-fold} sumset and {Subsequence} sum},
     journal = {Comptes Rendus. Math\'ematique},
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Mohan; Ram Krishna Pandey. Generalized H-fold sumset and Subsequence sum. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1-19. doi : 10.5802/crmath.483. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.483/

Bibliographie
Cité par

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[2] Jagannath Bhanja A note on sumsets and restricted sumsets, J. Integer Seq., Volume 24 (2021) no. 4, 21.4.2, 9 pages | MR | Zbl

[3] Jagannath Bhanja On the minimum cardinality of generalized sumsets in finite cyclic groups, Integers, Volume 21 (2021), A8, 16 pages | MR | Zbl

[4] Jagannath Bhanja; Ram Krishna Pandey Inverse problems for certain subsequence sums in integers, Discrete Math., Volume 343 (2020) no. 12, 112148, 11 pages | MR | Zbl

[5] Jagannath Bhanja; Ram Krishna Pandey On the minimum size of subset and subsequence sums in integers, C. R. Math. Acad. Sci. Paris, Volume 360 (2022), pp. 1099-1111 | MR | Zbl

[6] Raj Kumar Mistri; Ram Krishna Pandey A generalization of sumsets of sets of integers, J. Number Theory, Volume 143 (2014), pp. 334-356 | DOI | MR | Zbl

[7] Raj Kumar Mistri; Ram Krishna Pandey; Om Prakash Subsequence sums: Direct and inverse problems, J. Number Theory, Volume 148 (2015), pp. 235-256 | DOI | MR | Zbl

[8] Francesco Monopoli A generalization of sumsets modulo a prime, J. Number Theory, Volume 157 (2015), pp. 271-279 | DOI | MR | Zbl

[9] Melvyn B. Nathanson Inverse theorems for subset sums, Trans. Am. Math. Soc., Volume 347 (1995) no. 4, pp. 1409-1418 | DOI | MR | Zbl

[10] Melvyn B. Nathanson Additive Number Theory: inverse problems and the geometry of sumsets, Graduate Texts in Mathematics, 165, Springer, 1996

[11] Quan-Hui Yang; Yong-Gao Chen On the cardinality of general $h$ -fold sumsets, Eur. J. Comb., Volume 47 (2015), pp. 103-114 | DOI | MR | Zbl

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