Inverse problems for certain subsequence sums in integers

TL;DR

This paper investigates the minimal size and structure of sum sets derived from subsets and subsequences of integers, extending recent finite field results to the integer group and generalizing to subsequence sums.

Contribution

It determines the minimal cardinality of subset and subsequence sum sets in integers and characterizes the structure of extremal sets, generalizing recent finite field findings.

Findings

Established the minimum size of subset sum sets in integers.

Characterized the structure of sets achieving minimal sum set size.

Extended results to subsequence sums and recovered known special cases.

Abstract

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha} (A):=\{s(B): B \subset A, |B|\geq \alpha\}.\] Now, let $\mathcal{A}=(\underbrace{a_{1},\ldots,a_{1}}_{r_{1}~\text{copies}}, \underbrace{a_{2},\ldots,a_{2}}_{r_{2}~\text{copies}},\ldots, \underbrace{a_{k},\ldots,a_{k}}_{r_{k}~\text{copies}})$ be a finite sequence of integers with $k$ distinct terms, where $r_{i}\geq 1$ for $i=1,2,\ldots,k$. Given a subsequence $\mathcal{B}$ of $\mathcal{A}$, the sum of all terms of $\mathcal{B}$, denoted by $s(\mathcal{B})$, is called the subsequence sum of $\mathcal{B}$. For $0\leq \alpha \leq \sum_{i=1}^{k} r_{i}$, let \[\Sigma_{\alpha} (\bar{r},\mathcal{A}):=\left\{s(\mathcal{B}): \mathcal{B}~\text{is a subsequence of}~\mathcal{A}~\text{of length} \geq \alpha \right\},\] where $\bar{r}=(r_{1},r_{2},\ldots,r_{k})$. Very recently, Balandraud obtained the minimum cardinality of $\Sigma_{\alpha} (A)$ in finite fields. Motivated by Baladraud's work, we find the minimum cardinality of $\Sigma_{\alpha}(A)$ in the group of integers. We also determine the structure of the finite set $A$ of integers for which $|\Sigma_{\alpha} (A)|$ is minimal. Furthermore, we generalize these results of subset sums to the subsequence sums $\Sigma_{\alpha} (\bar{r},\mathcal{A})$. As special cases of our results we obtain some already known results for the usual subset and subsequence sums.