On the minimum size of subset and subsequence sums in integers

Abstract

Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$. For a nonnegative integer $\alpha \le rk$, let $\Sigma _{\alpha } (\mathcal{A})$ be the set of all subsequence sums of $\mathcal{A}$ that correspond to the subsequences of length $\alpha $ or more. When $r=1$, we call the subsequence sums as subset sums and we write $\Sigma _{\alpha } (A)$ for $\Sigma _{\alpha } (\mathcal{A})$. In this article, using some simple combinatorial arguments, we establish optimal lower bounds for the size of $\Sigma _{\alpha } (A)$ and $\Sigma _{\alpha } (\mathcal{A})$. As special cases, we also obtain some already known results in this study.