Bounds on two-distance sets in Euclidean space and Unit Sphere

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. $$n\le {\displaystyle \frac{(d+1)({\left(\frac{1+{\delta }^2}{1-{\delta }^2}\right)}^2-1)}{{\left(\frac{1+{\delta }^2}{1-{\delta }^2}\right)}^2-(d+1)}}+1.$$ if the two distances are $1$ and $\delta$ in ${\mathbb{R}}^d$. For spherical two-distance sets with $n$ points and inner products $a,b$ on ${\mathbb{S}}^{d-1}$, we will have the following: $$\{\begin{array}{lll}n\le {\displaystyle \frac{d({\left({\displaystyle \frac{a+b-2}{b-a}}\right)}^2-1)}{{\left({\displaystyle \frac{a+b-2}{b-a}}\right)}^2-d}},& a+b\ge 0;n\le {\displaystyle \frac{(d+1)({\left({\displaystyle \frac{a+b-2}{b-a}}\right)}^2-1)}{{\left({\displaystyle \frac{a+b-2}{b-a}}\right)}^2-(d+1)}},& a+b<0.\end{array}$$ Notice that the second bound (for $a+b<0$) is the same as the relative bound for the equiangular lines in one higher dimension.