Year: 2025
Type: article
Abstract: We prove that the diameter of a Sidon set (also known as a Babcock sequence, Golomb ruler, or $$B_2$$ B 2 set) with $$k$$ k elements is at least $$k^2-b k^{3/2}-O(k)$$ k 2 - b k 3 / 2 - O ( k ) where $$b\le 1.96365$$ b ≤ 1.96365 , a comparatively large improvement on past results. Equivalently, a Sidon set with diameter $$n$$ n has at most $$n^{1/2}+0.98183n^{1/4}+O(1)$$ n 1 / 2 + 0.98183 n 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 $$b\le 1.99058$$ b ≤ 1.99058 that can be verified by hand, which still improves on past results. Finally, we prove that $$g$$ g -thin Sidon sets (aka $$g$$ g -Golomb rulers) with $$k$$ k elements have diameter at least $$g^{-1} k^2 - (2-\varepsilon)g^{-1}k^{3/2} - O(k)$$ g - 1 k 2 - ( 2 - ε ) g - 1 k 3 / 2 - O ( k ) , with $$\varepsilon\ge 0.0062g^{-4}$$ ε ≥ 0.0062 g - 4 .
Authors: D. Carter, Z. Hunter, K. O’Bryant
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4407231387