Sliding-window upper bound for Sidon sets ($|A|\le \sqrt{N}+O(N^{1/4})$)

For a Sidon set $A\subseteq [1,N]$, put $m=|A|$ and let $A_i$ count points in a window of $u$ consecutive integers. Each point occurs in $u$ windows, so $\sum A_i=um$. Cauchy-Schwarz gives $\sum \left(\genfrac{}{}{0ex}{}{A_i}{2}\right)\ge [u^2{m}^2/(N+u)-um]/2$. Each positive difference occurs at most once and contributes $u-d$ windows, giving the upper bound $u(u-1)/2$. Thus $m^2\le (N+u)(1+(m-1)/u)$. Taking $u=\lfloor N^{3/4}\rfloor$ gives the displayed estimate, which combines with the Bose-Chowla Sidon construction to determine the asymptotic maximum.