Generalized H-fold sumset and Subsequence sum

Mohan; Ram Krishna Pandey

doi:10.5802/crmath.483

Research article - Combinatorics, Number theory

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.

Abstract

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.

Received: 2022-09-07
Revised: 2023-02-21
Accepted: 2023-02-23
Published online: 2024-02-02

DOI: 10.5802/crmath.483

Classification: 11P70, 11B75, 11B13
Keywords: sumset, subset sum, subsequence sum

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Generalized {H-fold} sumset and {Subsequence} sum},
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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/

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Cited by

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