Density Frankl-R\"{o}dl on the Sphere

TL;DR

This paper proves a density version of the Frankl-R"{o}dl theorem on the sphere, establishing bounds on the probability of certain inner product configurations within large measure subsets.

Contribution

It introduces a density variant of the Frankl-R"{o}dl theorem on the sphere and extends to broader configurations, including inductive configurations and sphere Ramsey results.

Findings

Established lower bounds on probability for pairs in large measure subsets.

Proved density versions of spherical avoidance problems for various configurations.

Showed all inductive configurations are sphere Ramsey.

Abstract

We establish a density variant of the Frankl-R\"{o}dl theorem on the sphere $\mathbb{S}^{n-1}$, which concerns avoiding pairs of vectors with a specific distance, or equivalently, a prescribed inner product. In particular, we establish lower bounds on the probability that a randomly chosen pair of such vectors lies entirely within a measurable subset $A \subseteq \mathbb{S}^{n-1}$ of sufficiently large measure. Additionally, we prove a density version of spherical avoidance problems, which generalize from pairwise avoidance to broader configurations with prescribed pairwise inner products. Our framework encompasses a class of configurations we call inductive configurations, which include simplices with any prescribed inner product $-1 < r < 1$. As a consequence of our density statement, we show that all inductive configurations are sphere Ramsey.