{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-diameter-of-finite-sidon-sets","title":"On the Diameter of Finite Sidon Sets","arxiv_id":"2310.20032","date":"2023-10-30","proceeding":null,"authors":["Daniel Carter","Zach Hunter","Kevin O'Bryant"],"abstract":"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 $\\varepsilon\\ge 0.02g^{-2}$.","url_abs":"https://arxiv.org/abs/2310.20032v1","url_pdf":"https://arxiv.org/pdf/2310.20032v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-diameter-of-finite-sidon-sets","repo_url":"https://github.com/dcartermath/sidon","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}