On Additive Representation Functions

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