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

TL;DR

This paper extends algebraic methods to establish upper bounds on the size of s-distance sets within real algebraic sets, providing new proofs and generalizations of existing bounds using combinatorial and algebraic techniques.

Contribution

It introduces novel upper bounds for s-distance sets in various real algebraic sets, generalizing previous bounds and employing Gr"obner basis techniques.

Findings

Provided a new proof for the Delsarte-Goethals-Seidel bound on spherical s-distance sets.

Generalized the Bannai-Kawasaki-Nitamizu-Sato bound to unions of spheres.

Extended algebraic bounds to broader classes of real algebraic sets.

Abstract

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 $\mbox{$\cal A$}\subseteq {\mathbb R}^n$ 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-Kawasaki-Nitamizu-Sato on $s$-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Gr\"obner basis techniques.