Partitioning a cyclic group into well-dispersed subsets with a common gap sequence

Let k, and n be positive integers with n = k . Let a = {a1, a2, . . . , a } be a multiset of positive integers with a1 + a2 + · · ·+ a = n. Let Zn = [0, n− 1] be the cyclic group of order n. A partition of Zn into k parts T1, T2, . . . , Tk is called a well-dispersed partition with a gap sequence a , if (1) the cardinality of Ti∩[(j−1)k, jk− 1] is 1 for 1 ≤ i ≤ k, 1 ≤ j ≤ and (2) a set of distances of consecutive elements of Ti coincides with a as a multiset for 1 ≤ i ≤ k. Amongst other results, it is proved that if k is a power of 2, then there exists a well-dispersed partition with a gap sequence {1, 2, . . . , 2k − 1}.