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.
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THEME Logistics (theme 3)
Publisher Cornell University Library
Series arXiv.org e-Print archive
Project Spinoza prijs Lex Schrijver
Citation
de Oliveira Filho, F.M, & Vallentin, F. (2010). A quantitative version of Steinhaus’ theorem for compact, connected, rank-one symmetric spaces. arXiv.org e-Print archive. Cornell University Library .