Generalized H-fold sumset and Subsequence sum
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 $h$ element...
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Académie des sciences
2024-02-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.483/ |
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| author | Mohan Pandey, Ram Krishna |
| author_facet | Mohan Pandey, Ram Krishna |
| author_sort | Mohan |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-cb72ebb4919b486492554a2426054219 |
| 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-cb72ebb4919b486492554a24260542192025-02-07T11:12:53ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-02-01362G111910.5802/crmath.48310.5802/crmath.483Generalized H-fold sumset and Subsequence sumMohan0Pandey, Ram Krishna1Department of Mathematics, Indian Institute of Technology Roorkee, Uttarakhand 247667, IndiaDepartment of Mathematics, Indian Institute of Technology Roorkee, Uttarakhand, 247667, IndiaLet $A$ 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.483/sumsetsubset sumsubsequence sum |
| spellingShingle | Mohan Pandey, Ram Krishna Generalized H-fold sumset and Subsequence sum Comptes Rendus. Mathématique sumset subset sum subsequence sum |
| title | Generalized H-fold sumset and Subsequence sum |
| title_full | Generalized H-fold sumset and Subsequence sum |
| title_fullStr | Generalized H-fold sumset and Subsequence sum |
| title_full_unstemmed | Generalized H-fold sumset and Subsequence sum |
| title_short | Generalized H-fold sumset and Subsequence sum |
| title_sort | generalized h fold sumset and subsequence sum |
| topic | sumset subset sum subsequence sum |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.483/ |
| work_keys_str_mv | AT mohan generalizedhfoldsumsetandsubsequencesum AT pandeyramkrishna generalizedhfoldsumsetandsubsequencesum |