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

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 sev