K-complementing Subsets of Nonnegative Integers

Let S = {S1,S2, . . .} represent a collection of nonempty sets of nonnegative integers in which each member contains the integer 0. Then S is called a complementing system of subsets for X ⊆ {0,1, . . .} if every x ∈ X can be uniquely represented as x = s1 + s2 + ··· with si ∈ Si. We will also write X = S1 ⊕ S2 ⊕ ··· and, when necessary, refer to X as the direct sum of the Si. We will denote the set of all complementing systems for X by CS(X). Then {X} ∈ CS(X) = ∅. If there is a positive integer k such that X = S1 ⊕ ··· ⊕ Sk, then {S1, . . . ,Sk} will be called a k-complementing system of subsets, or a complementing k-tuple, for X . Denote the set of all complementing k-tuples for X by CS(k,X). We will address the problem of characterizing all S∈ CS(k,Nn), where Nn = {0,1, . . . , n− 1}. The corresponding more general problem for CS(N) was solved by de Bruijn [2], where N= {0,1, . . .}. Long [4] has given a complete solution for CS(2,Nn). Since the appearance of Long’s paper, no progress seems to have been made to solve the problem for k > 2. Tijdeman [6] gives a survey of the evolution of this problem and related work. In Section 2, we give an alternative proof of Long’s theorem (Theorem 2.5) followed in Section 3 by its natural extension (Theorem 3.2) and a general structure theorem for CS(k,Nn) (Theorem 3.5). A complementing system S = {S1,S2, . . .} ∈ CS(N) will be called usual if for any sequence g1,g2, . . . (gi > 1) of integers, each Si ∈ S is given by Si = { 0,mi−1,2mi−1, . . . , ( gi− 1 ) mi−1 } , (1.1)