Improved bounds for zero-sum cycles in $\mathbb{Z}_p^d$

TL;DR

This paper improves bounds on the minimum size of complete digraphs needed to guarantee zero-sum cycles in finite Abelian groups, specifically solving the case for $ ext{Z}_2^d$ and tightening bounds for other prime powers.

Contribution

It provides new tight bounds for $n( ext{Z}_p^d)$, especially solving the case for $p=2$, and introduces a generalized hypergraph matching result in a matroidal setting.

Findings

Established $n( ext{Z}_2^d) o 5d$ bound.

Proved $n( ext{Z}_p^d) o O(pd ext{log} d)$ for primes $p eq 2$.

Connected bounds to a conjecture on additive bases in $ ext{Z}_p^d$.

Abstract

For a finite Abelian group $(\Gamma,+)$, let $n(\Gamma)$ denote the smallest positive integer $n$ such that for each labelling of the arcs of the complete digraph of order $n$ using elements from $\Gamma$, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals $0$. Alon and Krivelevich initiated the study of the parameter $n(\cdot)$ on cyclic groups and proved that $n(\mathbb{Z}_q)=O(q\log q)$. Studying the prototypical case when $\Gamma=\mathbb{Z}_p^d$ is a power of a cyclic group of prime order, Letzter and Morrison recently showed that $n(\mathbb{Z}_p^d) \le O(pd(\log d)^2)$ and that $n(\mathbb{Z}_2^d)\le O(d \log d)$. They then posed the problem of proving an (asymptotically optimal) upper bound of $n(\mathbb{Z}_p^d)\le O(pd)$ for all primes $p$ and $d \in \mathbb{N}$. In this paper, we solve this problem for $p=2$ and improve their bound for all primes $p \ge 3$ by proving $n(\mathbb{Z}_2^d)\le 5d$ and $n(\mathbb{Z}_p^d)\le O(pd\log d)$. While the first bound determines $n(\mathbb{Z}_2^d)$ up to a multiplicative error of $5$, the second bound is tight up to a $\log d$ factor. Moreover, our result shows that a tight bound of $n(\mathbb{Z}_p^d)=\Theta(pd)$ for arbitrary $p$ and $d$ would follow from a (strong form) of the well-known conjecture of Jaeger, Linial, Payan and Tarsi on additive bases in $\mathbb{Z}_p^d$. Along the way to proving these results, we establish a generalization of a hypergraph matching result by Haxell in a matroidal setting. Concretely, we obtain sufficient conditions for the existence of matchings in a hypergraph whose hyperedges are labelled by the elements of a matroid, with the property that the edges in the matching induce a basis of the matroid. We believe that these statements are of independent interest.