Proofs and disproofs of some conjectures on zero-sum problems of new types

Abstract

For integers $r\ge 1$ and $n\ge 2$, let $s_r(n)$ be the least $N$ such that every sequence of $N$ vectors in ${\mathbb{Z}}^r$, each nonzero modulo $n$, contains $n$ terms whose sum is zero modulo $n$ but nonzero modulo $n^2$. Let $w_r(n)$ be the corresponding least $N$ when any nonempty subsequence is allowed. These constants were introduced as zero-sum problems of new types by Z.-W. Sun. We prove $s_r(n)\ge (2^{r+1}-2)(n-1)+1$ by constructing an explicit sequence from a lattice polytope, \emph{disproving} his conjecture $s_2(n)=4n+1$ for $n\ge 4$. We obtain $w_r(2)=2r+1$ by classifying all sequences avoiding such nonempty subsequences modulo $4$. We also \emph{prove} Sun's conjecture $w_2(n)=4n-3$ for $n=2^k{3}^{\ell }p$, where $p$ is a prime and $k,\ell \ge 0$ are integers, by classifying the extremal sequences of length $4p-4$ modulo $p^2$ when $n=p$. For every fixed $n\ge 2$, we prove $w_r(n)/s({\mathbb{Z}}_n^r)=O_n(r^2/2^r)\to 0$ as $r\to \infty$, where $s({\mathbb{Z}}_n^r)$ is the classical Erdős-Ginzburg-Ziv constant, with an explicit threshold on $r$ for strict inequality. Thus, for every fixed modulus $n\ge 2$, Sun's proposed equality fails for sufficiently large $r$.