2025/08/31 by Wei-Chun Chen, Chen, Wei-Chun, Wei-Hsuan Yu +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2509.00858
openalex publication_date 2025/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish upper bounds for the size of two-distance sets in Euclidean space and spherical two-distance sets. The main recipe for obtaining upper bounds is the spectral method. We construct Seidel matrices to encode the distance relations and apply eigenvalue analysis to obtain explicit bounds. For Euclidean space, we have the upper bounds for the cardinality n of a two-distance set. n ≤ \dfrac(d+1)(((1+δ2)/(1-δ2))2 - 1)((1+δ2)/(1-δ2))2-(d+1)+1. if the two distances are 1 and δ in ℝd. For spherical two-distance sets with n points and inner products a, b on \mathbbSd-1, we will have the following: \begincases n ≤ \dfracd((\dfraca+b-2b-a)2-1)(\dfraca+b-2b-a)2-d, amp;a+b ≥ 0; n ≤ \dfrac(d+1)((\dfraca+b-2b-a)2-1)(\dfraca+b-2b-a)2-(d+1), amp;a+b lt; 0. \endcases Notice that the second bound (for a+b < 0) is the same as the relative bound for the equiangular lines in one higher dimension.