On Additive Representation Functions

Abstract: Let $\A={a_1<a_2<a_3.....<a_n<...}$ be an infinite sequence of integers and let $R_2(n)=|{(i,j):\ \ a_i+a_j=n;\ \ a_i,a_j\in \A;\ \ i\le j}|$. We define $S_k=\s_{l=1}^k(R_2(2l)-R_2(2l+1))$. We prove that, if $L^{\infty}$ norm of $S_k^+(=\max{S_k,0})$ is small then $L^1$ norm of $\frac{S_k^+}{k}$ is large.