2024/01/16 by Mohan, Jagannath Bhanja, Bhanja, Jagannath +2
Mathematics · #11B13 #11B75 #11P70 #Advanced Topology and Set Theory #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2401.08208
openalex publication_date 2024/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a nonempty finite set A of integers, let S(A) = \ ∑b∈ B b: ∅ \not= B⊆ A\ be the set of all nonempty subset sums of A. In 1995, Nathanson determined the minimum cardinality of S(A) in terms of |A| and described the structure of A for which |S(A)| is the minimum. He asked to characterize the underlying set A if |S(A)| is a small increment to its minimum size. Problems of such nature are inspired by the well-known Freiman's 3k-4 theorem. In this paper, some results in the direction of Freiman's 3k-4 theorem for the set of subset sums S(A) are proved. Such results are also extended to the set of subsequence sums S(\mathbbA) = \ ∑_b∈ \mathbbB b: ∅ \not= \mathbbB ⊆ \mathbbA \ of sequence \mathbbA, where the notation \mathbbB ⊆ \mathbbA , is used for \mathbbB is a subsequence of \mathbbA. The results are further generalized to a generalization of subset and subsequence sums. The main idea of the proofs of the results is to write the set of subset sums S(A) and the set of subsequence sums S(\mathbbA) in terms of the h-fold sumset hA and the h-fold restricted sumset h^\wedge A. Such representation also gives other proof of some of the results of Nathanson and Mistri et al.