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 $\mathcal{A}\subseteq \mathbb{R}^n$ 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 Gröbner basis techniques.
@article{10_37236_9712,
author = {G\'abor Heged\"us and Lajos R\'onyai},
title = {An upper bound for the size of \(s\)-distance sets in real algebraic sets},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {3},
doi = {10.37236/9712},
zbl = {1467.05267},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9712/}
}
TY - JOUR
AU - Gábor Hegedüs
AU - Lajos Rónyai
TI - An upper bound for the size of \(s\)-distance sets in real algebraic sets
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/9712/
DO - 10.37236/9712
ID - 10_37236_9712
ER -
%0 Journal Article
%A Gábor Hegedüs
%A Lajos Rónyai
%T An upper bound for the size of \(s\)-distance sets in real algebraic sets
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/9712/
%R 10.37236/9712
%F 10_37236_9712
Gábor Hegedüs; Lajos Rónyai. An upper bound for the size of \(s\)-distance sets in real algebraic sets. The electronic journal of combinatorics, Tome 28 (2021) no. 3. doi: 10.37236/9712
