Let $A$ and $H$ be nonempty finite sets of integers and positive integers, respectively. The generalized $H$-fold sumset, denoted by ${H}^{\left(r\right)}A$, is the union of the sumsets ${h}^{\left(r\right)}A$ for $h\in H$ where, the sumset ${h}^{\left(r\right)}A$ 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 ${H}^{\left(r\right)}A$, i.e., for $|{H}^{\left(r\right)}A|$ and the structure of the underlying sets $A$ and $H$ when $|{H}^{\left(r\right)}A|$ 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 ${H}^{\left(r\right)}A$ generalizes subsequence sum and hence subset sum, we get several results of subsequence sums and subset sums as special cases.

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Keywords: sumset, subset sum, subsequence sum

Mohan ^{1};
Ram Krishna Pandey ^{2}

@article{CRMATH_2024__362_G1_1_0, author = {Mohan and Ram Krishna Pandey}, title = {Generalized {H-fold} sumset and {Subsequence} sum}, journal = {Comptes Rendus. Math\'ematique}, pages = {1--19}, publisher = {Acad\'emie des sciences, Paris}, volume = {362}, year = {2024}, doi = {10.5802/crmath.483}, language = {en}, }

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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