A quantitative version of Steinhaus' theorem for compact, connected, rank-one symmetric spaces

TL;DR

This paper provides a quantitative extension of Steinhaus' theorem for compact, connected, rank-one symmetric spaces, demonstrating exponential decay in measure for subsets avoiding specific distances.

Contribution

It introduces a method to select distances in symmetric spaces that ensures measure decay, extending Steinhaus' theorem quantitatively.

Findings

Measure of subsets avoiding certain distances decays exponentially.

Explicit construction of distances for measure control.

Generalization of Steinhaus' theorem to rank-one symmetric spaces.

Abstract

Let $d_1$, $d_2$, ... be a sequence of positive numbers that converges to zero. A generalization of Steinhaus' theorem due to Weil implies that, if a subset of a homogeneous Riemannian manifold has no pair of points at distances $d_1$, $d_2$, ... from each other, then it has to have measure zero. We present a quantitative version of this result for compact, connected, rank-one symmetric spaces, by showing how to choose distances so that the measure of a subset not containing pairs of points at these distances decays exponentially in the number of distances.