A converse to the Erdős-Fuchs theorem

Abstract

We prove that there exists $A\subseteq \mathbb{N}$ such that \[ R_A(N)=\fracπ{4}N+ O\!\left(N^{1/4}\sqrt{\log N}\right), \] where $R_A(N)=\#\{(a,b)\in A^2:a+b\le N\}$. This improves the record $O(N^{1/4}\log N)$ obtained by Ruzsa in 1997.