Pinned Geometric Configurations in Euclidean Space and Riemannian Manifolds

Alex Iosevich, Krystal Taylor and Ignacio Uriarte-Tuero
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Alex Iosevich: Department of Mathematics, University of Rochester, P.O. Box 270138, Rochester, NY 14627, USA
Krystal Taylor: Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA
Ignacio Uriarte-Tuero: Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada

Mathematics, 2021, vol. 9, issue 15, 1-17

Abstract: Let M be a compact d -dimensional Riemannian manifold without a boundary. Given a compact set E ? M , we study the set of distances from the set E to a fixed point x ? E . This set is ? ? x ( E ) = { ? ( x , y ) : y ? E } , where ? is the Riemannian metric on M . We prove that if the Hausdorff dimension of E is greater than d + 1 2 , then there exist many x ? E such that the Lebesgue measure of ? ? x ( E ) is positive. This result was previously established by Peres and Schlag in the Euclidean setting. We give a simple proof of the Peres–Schlag result and generalize it to a wide range of distance type functions. Moreover, we extend our result to the setting of chains studied in our previous work and obtain a pinned estimate in this context.

Keywords: pinned distance sets; Falconer conjecture; Riemannian manifolds; fractals (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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