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arXiv · 2609.27673

A converse to the Erdős-Fuchs theorem

Quan-Hui Yang·Lilu Zhao
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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.

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Quan-Hui Yang, Lilu Zhao. 2026-09-23. A converse to the Erdős-Fuchs theorem. https://arxiv.org/abs/2609.27673

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