On the minimum size of subset and subsequence sums in integers

TL;DR

This paper establishes optimal lower bounds on the number of distinct subset and subsequence sums in integer sequences with specific multiplicity constraints, generalizing and recovering known results in additive combinatorics.

Contribution

It provides new combinatorial bounds for the size of subsequence sum sets in sequences with repeated elements, extending previous results.

Findings

Derived optimal lower bounds for subsequence sum sets.

Unified and generalized existing results in additive combinatorics.

Applied combinatorial arguments to establish bounds.

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 $\alpha \leq rk$, 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.