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Pinned Geometric Configurations in Euclidean Space and Riemannian Manifolds

Author

Listed:
  • Alex Iosevich

    (Department of Mathematics, University of Rochester, P.O. Box 270138, Rochester, NY 14627, USA
    Current address: 100 Math Tower, 231 W 18th Ave, Columbus, OH 43210, USA.
    These authors contributed equally to this work.)

  • Krystal Taylor

    (Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA
    These authors contributed equally to this work.)

  • Ignacio Uriarte-Tuero

    (Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
    These authors contributed equally to this work.)

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 Δ ρ x ( E ) = { ρ ( x , y ) : y ∈ E } , where ρ 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 Δ ρ 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.

Suggested Citation

  • Alex Iosevich & Krystal Taylor & Ignacio Uriarte-Tuero, 2021. "Pinned Geometric Configurations in Euclidean Space and Riemannian Manifolds," Mathematics, MDPI, vol. 9(15), pages 1-17, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1802-:d:604232
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