ON PARTITIONS OF NONNEGATIVE INTEGERS AND REPRESENTATION FUNCTIONS

Let N be the set of all nonnegative integers. For any set A subset of N, let R(A, n) denote the number of representations of n as n = a + a' with a, a' is an element of A. There is no partition N = A boolean OR B such that R(A, n) = R(B, n) for all sufficiently large integers n. We prove that a partition N = A boolean OR B satisfies vertical bar R(A, n) - R(B, n)vertical bar <= 1 for all nonnegative integers n if and only if, for each nonnegative integer m, exactly one of 2m + 1 and 2m is in A.