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

Fernando Mário de Oliveira Filho; Frank Vallentin

doi:10.1007/s10711-012-9814-1

paper

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

arXiv:1005.0471 · doi:10.1007/s10711-012-9814-1

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.

12 pages, 1 figure. Accepted at Geometriae Dedicata

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

Abstract

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