A note on the lower bound of representation functions

For a set A of nonnegative integers, let R-2(A,n) denote the number of solutions to n = a + a ' with a,a 'is an element of A, a < a '. Let A(0) be the Thue-Morse sequence and B-0 = N\A(0). Let A subset of N and N be a positive integer such that R-2(A,n) = R-2(N\A,n) for all n >= 2N - 1. Previously, the first author proved that if |A boolean AND A(0)| = +infinity and |A boolean AND B-0| = +infinity, then R-2(A,n) >= n+3/56N-52 - 1 for all n >= 1. In this paper, we prove that the above lower bound is nearly best possible. We also get some other results.