An upper bound for the size of s-distance sets in real algebraic sets

In a recent paper, Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester's Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of s-distance sets (subsets A subset of R-n which determine at most s different distances). In this paper we extend their work and prove upper bounds for the size of s-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-Goethals-Seidel on spherical s-distance sets and a generalization of a bound by Bannai-KawasakiNitamizu-Sato on s-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Grobner basis techniques.