Some New Results on Purely Singular Splittings (Communications in Mathematical Research, 2022)
Let $G$ be a finite abelian group, $M$ a set of integers and $S$ a subsetof $G$. 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 hasno such representation. The splitting is called purely singular if for each primedivisor $p$ of $|G|$, there is at least one element of $M$ is divisible by $p$. In thispaper, we continue the study of purely singular splittings of cyclic groups. Weprove that if $k≥ 2$ is a positive integer such that $[−2k+1, 2k+2]^∗$ splits a cyclicgroup $\mathbb{Z}_m$, then $m=4k+2$. We prove also that if $M=[−k_1, k_2]^∗$ splits $\mathbb{Z}_m$ purelysingularly, and $15 ≤ k_1+k_2 ≤ 30$, then $m = 1$, or $m = k_1+k_2+1$, or $k_1 = 0$ and $m=2k_2+1$.