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An Improved Upper Bound for Finite Sidon Sets via Vector-Valued Smoothing

  • Jianfeng Hou
  • Hongbin Zhao
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Abstract

Let F(N) denote the largest cardinality of a Sidon subset of {0,1,…,N−1}. We prove F(N)≤ N1/2+γ0N1/4+O(1), γ0=0.94349…<0.9435. This improves the previously published coefficient $0.98183$. Our argument develops a vector-valued smoothing method that combines several discrete smoothing kernels, each accompanied by boundary weights that compensate for endpoint effects, so that their weighted combination satisfies the required finite covering inequalities. We also show that averaging systems that satisfy these inequalities individually cannot improve upon the best constituent. Numerical optimization is used to find an eight-component candidate system, which is then certified by exact rational arithmetic.

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Cite

BibTeXRIS
BibTeX
@misc{hou2026improved,
  title = {{An Improved Upper Bound for Finite Sidon Sets via Vector-Valued Smoothing}},
  author = {Jianfeng Hou and Hongbin Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.01169},
  eprint = {2607.01169},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.01169},
}
RIS
TY  - UNPB
TI  - An Improved Upper Bound for Finite Sidon Sets via Vector-Valued Smoothing
AU  - Jianfeng Hou
AU  - Hongbin Zhao
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2607.01169
UR  - https://arxiv.org/abs/2607.01169
AB  - Let F(N) denote the largest cardinality of a Sidon subset of {0,1,…,N−1}. We prove F(N)≤ N^1/2+γ_0N^1/4+O(1), γ_0=0.94349…<0.9435. This improves the previously published coefficient $0.98183$. Our argument develops a vector-valued smoothing method that combines several discrete smoothing kernels, each accompanied by boundary weights that compensate for endpoint effects, so that their weighted combination satisfies the required finite covering inequalities. We also show that averaging systems that satisfy these inequalities individually cannot improve upon the best constituent. Numerical optimization is used to find an eight-component candidate system, which is then certified by exact rational arithmetic.
ER  -