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arXiv · 2607.01169

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 $\{0,1,\ldots,N-1\}$. We prove \[ F(N)\le N^{1/2}+\gamma_0N^{1/4}+O(1), \qquad \gamma_0=0.94349\ldots<0.9435. \] 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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Jianfeng Hou, Hongbin Zhao. 2026-07-01. An Improved Upper Bound for Finite Sidon Sets via Vector-Valued Smoothing. https://arxiv.org/abs/2607.01169

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