发表机构
Fuzhou University（福州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
通过向量值卷积不等式和精确有理数验证，将Sidon集最大基数上界改进为N^{1/2} + 0.94601 N^{1/4} + O(1)。
AI 中文摘要
设$F(N)$表示$\{0, 1, \dots, N - 1\}$中Sidon子集的最大基数。我们证明\\[ F(N) \le N^{1/2} + 0.94601 N^{1/4} + O(1). \\] 这改进了最近由Carter、Georgiev、Gómez-Serrano、Hunter、O'Bryant、Tao和Wagner得到的系数$0.97633$。它也非常接近并数值上低于暂报值约$0.947$。论证基于一个向量值卷积不等式：几个平滑核共同承担产生边界优函数的任务，同时它们的$L^2$能量被平均。解析简化是初等的。最终常数由一个有限有理证书提供，该证书通过仅使用精确算术的短程序验证。
英文摘要
Let $F(N)$ denote the largest cardinality of a Sidon subset of $\{0,1,\ldots,N-1\}$. We prove \[ F(N)\le N^{1/2}+γ_0N^{1/4}+O(1), \qquad γ_0=0.94349\ldots<0.9435. \] 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.
Comments19 pages
