Inverse problems for certain subsequence sums in integers

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 $\alpha$ ($\le k$), let \[Σ_α (A):=\{s(B): B \subset A, |B|\geq α\}.\] Now, let $\mathcal{A}=(\underset{r_1\text{copies}}{\underbrace{a_1,\dots ,a_1}},\underset{r_2\text{copies}}{\underbrace{a_2,\dots ,a_2}},\dots ,\underset{r_k\text{copies}}{\underbrace{a_k,\dots ,a_k}})$ be a finite sequence of integers with $k$ distinct terms, where $r_i\ge 1$ for $i=1,2,\dots ,k$. 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 $0\le \alpha \le {\sum }_{i=1}^k{r}_i$, let \[Σ_α (\bar{r},\mathcal{A}):=\left\{s(\mathcal{B}): \mathcal{B}~\text{is a subsequence of}~\mathcal{A}~\text{of length} \geq α\right\},\] where $\bar{r}=(r_1,r_2,\dots ,r_k)$. 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.

15 pages