A quantitative version of Steinhaus%26apos; theorem for compact, connected, rank-one symmetric spaces

作者：de Oliveira Filho Fernando Mario^*; Vallentin Frank

来源：Geometriae Dedicata, 2013, 167(1): 295-307.

DOI：10.1007/s10711-012-9814-1

• Steinhaus%26apos
• theorem
• Geometric Ramsey theory
• Linear programming
• Orthogonal polynomials

Let , ... be a sequence of positive numbers that converges to zero. A generalization of Steinhaus%26apos; theorem due to Weil implies that, if a subset of a homogeneous Riemannian manifold has no pair of points at distances , ... 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.

• 出版日期2013-12