Raul Marquez (University of Texas Rio Grande Valley)「Analyzing bounds on $s$-distance sets utilizing distance relations」
Consider a finite set $\mathcal{C} = \{ x_1, x_2, \dots, x_n \}$ in a metric space, $X$. It is considered an $s$-distance set if the set of distances between distinct points a set of size $s$, denoted as $(d_1,d_2, \dots, d_s)$. Nozaki proved a general bound on these sets with Spherical codes and it was expanded by Musin and Barg to compact distance-transitive spaces. The method relies on a family of orthogonal polynomials associated with the compact distance-transitive space and represents the annihilator polynomial $\prod (d_i - x)$ in its basis. The bound connects these positive coefficients to the size of $C$. The presentation covers cases in which the annihilator polynomial has degree higher than $s$, examples of cases satisfying the bound, and considers various metric spaces.