TY - JOUR
T1 - Some New Results on Purely Singular Splittings
AU - Yuan , Pingzhi
JO - Communications in Mathematical Research 
VL - 2
SP - 136
EP - 156
PY - 2022
DA - 2022/02
SN - 38
DO - http://doi.org/10.4208/cmr.2020-0048
UR - https://global-sci.org/intro/article_detail/cmr/20268.html
KW - Splitter sets, perfect codes, factorizations of cyclic groups.
AB - <p style="text-align: justify;">Let $G$ be a finite abelian group, $M$ a set of integers and $S$ a subset
of $G$. We say that $M$ and $S$ form a splitting of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$, while 0 has
no such representation. The splitting is called purely singular if for each prime
divisor $p$ of $|G|$, there is at least one element of $M$ is divisible by $p$. In this
paper, we continue the study of purely singular splittings of cyclic groups. We
prove that if $k\geq 2$ is a positive integer such that $[−2k+1, 2k+2]^∗$ splits a cyclic
group $\mathbb{Z}_m$, then $m=4k+2$. We prove also that if $M=[−k_1, k_2]^∗$ splits $\mathbb{Z}_m$ purely
singularly, and $15 \leq k_1+k_2 \leq 30$, then $m = 1$, or $m = k_1+k_2+1$, or $k_1 = 0$ and $m=2k_2+1$.</p>