# On complete subsets of the cyclic group

## TL;DR

This paper proves a conjecture by Vu that a subset of the cyclic group Z_n with size greater than 1+2√(n-4) is complete, meaning every element of the subgroup can be expressed as a sum of distinct elements from the subset.

## Contribution

The paper establishes the exact threshold size for completeness of subsets in cyclic groups, confirming Vu's conjecture and extending previous results.

## Key findings

- Proves that subsets larger than 1+2√(n-4) are complete in Z_n.
- Confirms Vu's conjecture on the size threshold for completeness.
- Extends Olson's and Erdős-Heilbronn's results to a more general setting.

## Abstract

A subset $X$ of an abelian $G$ is said to be {\em complete} if every element of the subgroup generated by $X$ can be expressed as a nonempty sum of distinct elements from $X$. Let $A\subset \Z_n$ be such that all the elements of $A$ are coprime with $n$. Solving a conjecture of Erd\H{o}s and Heilbronn, Olson proved that $A$ is complete if $n$ is a prime and if $|A|>2\sqrt{n}.$ Recently Vu proved that there is an absolute constant $c$, such that for an arbitrary large $n$, $A$ is complete if $|A|\ge c\sqrt{n},$ and conjectured that 2 is essentially the right value of $c$. We show that $A$ is complete if $|A|> 1+2\sqrt{n-4}$, thus proving the last conjecture.