Freiman's $(3k-4)$-like results for subset and subsequence sums

Abstract: For a nonempty finite set $A$ of integers, let $S(A) = \left{ \sum_{b\in B} b: \emptyset \not= B\subseteq A\right}$ 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 $S(\mathbb{A}) = \left{ \sum_{b\in \mathbb{B}} b: \emptyset \not= \mathbb{B} \subseteq \mathbb{A} \right}$ of sequence $\mathbb{A}$, where the notation $\mathbb{B} \subseteq \mathbb{A} $, is used for $\mathbb{B}$ is a subsequence of $\mathbb{A}$. 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 $S(\mathbb{A})$ 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.