A zero-sum problem related to the max gap of the unit group of the residue class ring

Let S be a sequence over a finite abelian group G and $v_g\left(S\right)$ be the times that $g\in G$ occurs in S. A sequence S over G is called weak-regular if $v_g\left(S\right)\le \mathrm{ord}\left(g\right)$ for every $g\in G$. Denote by $N\left(G\right)$ the smallest integer t such that every weak-regular sequence S over G of length $\left|S\right|\ge t$ has a nonempty zero-sum subsequence T of S satisfying $v_g\left(T\right)=v_g\left(S\right)$ for some $g|S$. $N\left(G\right)$ has been formulated by Gao et al. very recently to study zero-sum problems in a unify way and determined only for cyclic groups of prime-power order and some other very special groups. As for general cyclic groups $G=C_n$, they gave that$2n-\lceil 3\sqrt{n}\rceil +1\le N\left(G\right)\le 2n-\lceil 2\sqrt{n+1}\rceil +1.$

In this paper, we first study the max gap of the unit group of the residue class ring and give an upper bound of it. Then we prove that there is always an integer $a\in [n^{\frac{1}{2}},n^{\frac{1}{2}}+n^{\frac{1}{4}}]$ such that $\gcd (a,n)=1$ for $n\ge 2227$. Finally, we improve the result of Gao et al. by showing that$2n-\lceil 2\sqrt{n+1}\rceil \le N\left(G\right)\le 2n-\lceil 2\sqrt{n+1}\rceil +1$ for any cyclic group $G=C_n$ with $n\ge 3$, in which for each equality, there are infinitely many n making it hold. And a computing result prefigures that $N\left(G\right)$ has not been determined only for very few cyclic groups G.