@Article{CMR-38-2,
author = {Yuan, Pingzhi},
title = {Some New Results on Purely Singular Splittings},
journal = {Communications in Mathematical Research },
year = {2022},
volume = {38},
number = {2},
pages = {136--156},
abstract = {<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>},
issn = {2707-8523},
doi = {https://doi.org/10.4208/cmr.2020-0048},
url = {https://global-sci.com/article/81481/some-new-results-on-purely-singular-splittings}
}