  We prove that the diameter of a Sidon set (also known as a Babcock sequence,
Golomb ruler, or $B_2$ set) with $k$ elements is at least $k^2-b k^{3/2}-O(k)$
where $b\le 1.96365$, a comparatively large improvement on past results.
Equivalently, a Sidon set with diameter $n$ has at most
$n^{1/2}+0.98183n^{1/4}+O(1)$ elements. The proof is conceptually simple but
very computationally intensive, and the proof uses substantial computer
assistance. We also provide a proof of $b\le 1.99058$ that can be verified by
hand, which still improves on past results. Finally, we prove that $g$-thin
Sidon sets (aka $g$-Golomb rulers) with $k$ elements have diameter at least
$g^{-1} k^2 - (2-\varepsilon)g^{-1}k^{3/2} - O(k)$, with $\varepsilon\ge
0.02g^{-2}$.
