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PreprintCC BY 4.0arXiv

A converse to the Erdős-Fuchs theorem

  • Quan-Hui Yang
  • Lilu Zhao
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Abstract

We prove that there exists A⊆ ℕ such that RA(N)=π/4N+ O (N1/4√(log N)), where RA(N)=#{(a,b)∈ A2:a+b≤ N}. This improves the record O(N1/4log N) obtained by Ruzsa in 1997.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026converse,
  title = {{A converse to the Erdős-Fuchs theorem}},
  author = {Quan-Hui Yang and Lilu Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27673},
  eprint = {2609.27673},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27673},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A converse to the Erdős-Fuchs theorem
AU  - Quan-Hui Yang
AU  - Lilu Zhao
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27673
UR  - https://arxiv.org/abs/2609.27673
AB  - We prove that there exists A⊆ ℕ such that R_A(N)=π/4N+ O (N^1/4√(log N)), where R_A(N)=#{(a,b)∈ A^2:a+b≤ N}. This improves the record O(N^1/4log N) obtained by Ruzsa in 1997.
ER  -