A nonlinear bound for the number of subsequence sums

TL;DR

This paper establishes a lower bound on the number of distinct subsequence sums for zero-sum-free sequences over abelian groups, revealing a nonlinear relationship unless the sequence is structurally constrained.

Contribution

It introduces a nonlinear lower bound on subsequence sums for zero-sum-free sequences and characterizes sequences that do not meet this bound.

Findings

Zero-sum-free sequences have at least c|α|^{4/3} distinct subsequence sums.

Sequences with fewer sums are structurally controlled by few terms.

The bound is tight up to a constant factor.

Abstract

We show that a finite zero-sum-free sequence $\alpha$ over an abelian group has at least $c|\alpha|^{4/3}$ distinct subsequence sums, unless $\alpha$ is "controlled" by a small number of its terms; here $|\alpha|$ denotes the number of terms of $\alpha$, and $c>0$ is an absolute constant.