Papers › On the Diameter of Finite Sidon Sets
On the Diameter of Finite Sidon Sets
Daniel Carter, Zach Hunter, Kevin O'Bryant
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We prove that the diameter of a Sidon set (also known as a Babcock sequence, Golomb ruler, or B₂ set) with k elements is at least k²-b k^(3/2)-O(k) where b≤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≤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 k² - (2-ε)g⁻¹k^(3/2) - O(k), with ε≥0.02g⁻².
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