(untitled) (Communications in Mathematical Research, 2022)
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 mM and sS, 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≥ 2 is a positive integer such that [−2k+1, 2k+2]^∗ splits a cyclic group Z_m, then m=4k+2. We prove also that if M=[−k_1, k_2]^∗ splits Z_m purely singularly, 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.