On the diameter of finite Sidon sets

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 \(\varepsilon\ge 0.0062g^{-4}\).