Few distance sets in $\ell_p$ spaces and $\ell_p$ product spaces

TL;DR

This paper improves bounds on the maximum size of distance sets in $ ext{ell}_p$ spaces and their product spaces, extending results to large $p$, $s$-distance sets, and equilateral sets in various $ ext{ell}_p$ sums.

Contribution

It provides new upper bounds for distance and equilateral sets in $ ext{ell}_p$ spaces, generalizing previous results and covering a range of $p$ values and set types.

Findings

Improved upper bounds for $n+1$ points in $ ext{ell}_p$ spaces for large $p$

Generalized bounds to $s$-distance sets

Derived bounds for equilateral sets in $ ext{ell}_p$ sums

Abstract

Kusner asked if $n+1$ points is the maximum number of points in $\mathbb{R}^n$ such that the $\ell_p$ distance $(1<p<\infty)$ between any two points is $1$. We present an improvement to the best known upper bound when $p$ is large in terms of $n$, as well as a generalization of the bound to $s$-distance sets. We also study equilateral sets in the $\ell_p$ sums of Euclidean spaces, deriving upper bounds on the size of an equilateral set for when $p=\infty$, $p$ is even, and for any $1\le p<\infty$.