A converse to the Erdős-Fuchs theorem

We prove that there exists $A\subseteq \mathbb{N}$ such that $$R_A(N)=\frac{\pi }{4}N+O\text{\hspace{0.17em}⁣}\left(N^{1\mathrm{/}4}\sqrt{\log N}\right),$$ where $R_A(N)=\mathrm{\#}\{(a,b)\in A^2:a+b\le N\}$. This improves the record $O(N^{1\mathrm{/}4}\log N)$ obtained by Ruzsa in 1997.