The coset factorization of finite cyclic group

Let $G$ be a finite cyclic group, written additively, and let $A,B$ be nonempty subsets of $G$. We will say that $G=A+B$ is a $\mathit{\text{factorization}}$ if for each $g$ in $G$ there are unique elements $a,b$ of $G$ such that $g=a+b,a\in A,b\in B$. In particular, if $A$ is a complete set of residues $modulo$ $\mathrm{\mid }A\mathrm{\mid }$, then we call the factorization a $\mathit{\text{coset}}\mathit{\text{factorization}}$ of $G$. In this paper, we mainly study a factorization $G=A+B$, where $G$ is a finite cyclic group and $A=[0,n-k-1]\cup \{i_0,i_1,\dots i_{k-1}\}$ with $\mathrm{\mid }A\mathrm{\mid }=n$ and $n\ge 2k+1$. We obtain the following conclusion: If $(i)$ $k\leq 2$or$(ii)$Thenumberofdistinctprimedivisorsof$gcd(|A|,|B|)$isatmost$1$or$(iii)$ $gcd(\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid })=pq$ with $gcd(pq,\frac{\mathrm{\mid }B\mathrm{\mid }}{gcd(\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid })})=1$, then $A$ is a complete set of residues $modulo$ $n$.