2024/01/13 by Mohan, Ram Krishna Pandey, Pandey, Ram Krishna
Computer Science · Mathematics · #11B13 #11B75 #11P70 #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2401.07116
openalex publication_date 2024/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∈ 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.